Research ArticleEstimation of the Derivatives of a Function in a ConvolutionRegression Model with Random Design

Christophe Chesneau1 and Maher Kachour2

1Laboratoire de Mathematiques Nicolas Oresme Universite de Caen BP 5186 14032 Caen Cedex France2Ecole Superieure de Commerce IDRAC 47 rue Sergent Michel Berthet CP 607 69258 Lyon Cedex 09 France

Correspondence should be addressed to Christophe Chesneau christophechesneaugmailcom

Received 8 August 2014 Revised 25 February 2015 Accepted 5 March 2015

Academic Editor Jos De Brabanter

Copyright copy 2015 C Chesneau and M KachourThis is an open access article distributed under theCreative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

A convolution regression model with random design is considered We investigate the estimation of the derivatives of an unknownfunction element of the convolution product We introduce new estimators based on wavelet methods and provide theoreticalguarantees on their good performances

1 Introduction

We consider the convolution regression model with randomdesign described as follows Let (119884

1 119883

1) (119884

119899 119883

119899) be

119899 iid random variables defined on a probability space(ΩAP) where

119884V = (119891 ⋆ 119892) (119883V) + 120585V V = 1 119899 (1)

(119891 ⋆ 119892)(119909) = intinfin

minusinfin119891(119905)119892(119909 minus 119905)119889119905 119891 R rarr R is an unknown

function 119892 R rarr R is a known function 1198831 119883

119899are

119899 iid random variables with common density ℎ R rarr

[0infin) and 1205851 120585

119899are 119899 iid random variables such that

E(1205851) = 0 and E(1205852

1) lt infin Throughout this paper we assume

that 119891 119892 and ℎ are compactly supported with supp(119891) =[119886 119887] supp(119892) = [119886

1015840 1198871015840] supp(ℎ) = [119886lowast 119887lowast] (119886 119887 1198861015840 1198871015840) isin

R4 119886 lt 119887 1198861015840 lt 1198871015840 119886lowast= 119886 + 1198861015840 119887

lowast= 119887 + 1198871015840 119891 is 119898

times differentiable with119898 isin N 119892 is integrable and ordinarysmooth (the precise definition is given by (K2) in Section 31)and 119883V and 120585V are independent for any V = 1 119899 We aimto estimate the unknown function 119891 and its 119898th derivativedenoted by 119891(119898) from the sample (119884

1 119883

1) (119884

119899 119883

119899)

The motivation of this problem is the deconvolution ofa signal 119891 from 119891 ⋆ 119892 perturbed by noise and randomlyobserved The function 119892 can represent a driving force thatwas applied to a physical system Such situations naturallyappear in various applied areas as astronomy optics seismol-ogy and biology Model (1) can also be viewed as a natural

extension of some 1-periodic convolution regression modelsas those considered by for example Cavalier and Tsybakov[1] Pensky and Sapatinas [2] and Loubes and Marteau [3]In the form (1) it has been considered in Bissantz and Birke[4] and Birke et al [5] with a deterministic design andin Hildebrandt et al [6] with a random design These lastworks focus on kernelmethods and establish their asymptoticnormality The estimation of 119891(119898) more general to 119891 =

119891(0) is of interest to examine possible bumps and to studythe convexity-concavity properties of 119891 (see for instancePrakasa Rao [7] for standard statistical models)

In this paper we introduce new estimators for 119891(119898) basedon wavelet methods Through the use of a multiresolutionanalysis these methods enjoy local adaptivity against discon-tinuities and provide efficient estimators for a wide variety ofunknown functions119891(119898) Basics on wavelet estimation can befound in for example Antoniadis [8] Hardle et al [9] andVidakovic [10] Results on the wavelet estimation of 119891(119898) inother regression frameworks can be found in for exampleCai [11] Petsa and Sapatinas [12] and Chesneau [13]

The first part of the study is devoted to the case whereℎ the common density of 119883

1 119883

119899 is known We develop

a linear wavelet estimator and an adaptive nonlinear waveletestimator The second one uses the double hard thresholdingtechnique introduced byDelyon and Juditsky [14] It does notdepend on the smoothness of 119891(119898) in its construction it isadaptive We exhibit their rates of convergence via the mean

Hindawi Publishing CorporationAdvances in StatisticsVolume 2015 Article ID 695904 11 pageshttpdxdoiorg1011552015695904

2 Advances in Statistics

integrated squared error (MISE) and the assumption that119891(119898) belongs to Besov ballsTheobtained rates of convergencecoincide with existing results for the estimation of 119891(119898) in the1-periodic convolution regression models (see for instanceChesneau [15])

The second part is devoted to the case where ℎ isunknownWe construct a new linear wavelet estimator usinga plug-in approach for the estimation of ℎ Its constructionfollows the idea of the ldquoNES linear wavelet estimatorrdquo intro-duced by Pensky and Vidakovic [16] in another regressioncontext Then we investigate its MISE properties when 119891(119898)belongs to Besov balls which naturally depend on the MISEof the considered estimator for ℎ Furthermore let us men-tion that all our results are provedwith onlymoments of order2 on 120585

1 which provides another theoretical contribution to

the subjectThe remaining part of this paper is organized as fol-

lows In Section 2 we describe some basics on wavelets andBesov balls and present our wavelet estimation methodologySection 3 is devoted to our estimators and their perfor-mances The proofs are carried out in Section 4

2 Preliminaries

This section is devoted to the presentation of the consideredwavelet basis the Besov balls and our wavelet estimationmethodology

21 Wavelet Basis Let us briefly present the wavelet basis onthe interval [119886 119887] (119886 119887) isin R2 introduced by Cohen et al [17]Let 120601 and120595 be the initial wavelet functions of the Daubechieswavelets family db2N with 119873 ge 1 (see eg Daubechies[18])These functions have the distinction of being compactlysupported and belong to the class C119886 for 119873 gt 5119886 For any119895 ge 0 and 119896 isin Z we set

120601119895119896(119909) = 2

1198952120601 (2

119895119909 minus 119896)

120595119895119896(119909) = 2

1198952120595 (2

119895119909 minus 119896)

(2)

With appropriated treatments at the boundaries thereexist an integer 120591 and a set of consecutive integers Λ

119895of

cardinality proportional to 2119895 (both depending on 119886 119887 and119873) such that for any integer ℓ ge 120591

B = 120601ℓ119896 119896 isin Λ

ℓ 120595

119895119896 119895 isin N minus 0 ℓ minus 1 119896 isin Λ

119895

(3)

forms an orthonormal basis of the space of squared integrablefunctions on [119886 119887] that is

L2([119886 119887]) = 119906 [119886 119887] 997888rarr R (int

119887

119886

|119906 (119909)|2119889119909)

12

lt infin

(4)

For the case119886 = 0 and 119887 = 1 120591 is the smallest integer satisfying2120591 ge 2119873 and Λ

119895= 0 2119895 minus 1

For any integer ℓ ge 120591 and 119906 isin L2([119886 119887]) we have thefollowing wavelet expansion

119906 (119909) = sum119896isinΛ ℓ

119888ℓ119896120601ℓ119896(119909) +

infin

sum119895=ℓ

sum119896isinΛ 119895

119889119895119896120595119895119896(119909) 119909 isin [119886 119887]

(5)

where

119888119895119896= int

119887

119886

119906 (119909) 120601119895119896(119909) 119889119909

119889119895119896= int

119887

119886

119906 (119909) 120595119895119896(119909) 119889119909

(6)

An interesting feature of the wavelet basis is to providesparse representation of 119906 only few wavelet coefficients 119889

119895119896

characterized by a high magnitude reveal the main details of119906 See for example Cohen et al [17] and Mallat [19]

22 Besov Balls We say that a function 119906 isin L2([119886 119887]) belongsto the Besov ball 119861119904

119901119903(119872) with 119904 gt 0 119901 ge 1 119903 ge 1 and119872 gt 0

if there exists a constant119862 gt 0 such that 119888119895119896

and119889119895119896

(6) satisfy

2120591(12minus1119901)

( sum119896isinΛ 120591

|119888120591119896|119901)

1119901

+ (

infin

sum119895=120591

(2119895(119904+12minus1119901)

( sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816

119901

)

1119901

)

119903

)

1119903

le 119862

(7)

with the usual modifications if 119901 = infin or 119903 = infinThe interest of Besov balls is to contain various kinds

of homogeneous and inhomogeneous functions 119906 See forexample Meyer [20] Donoho et al [21] and Hardle et al [9]

23 Wavelet Estimation Let 119891 be the unknown function in(1) and B the considered wavelet basis taken with 119873 gt 5119898

(to ensure that 120601 and 120595 belong to the classC119898) Suppose that119891(119898) exists with 119891(119898) isin L2([119886 119887])Thefirst step in thewavelet estimation consists in expand-

ing 119891(119898) onB as

119891(119898)(119909) = sum

119896isinΛ ℓ

119888(119898)

ℓ119896120601ℓ119896(119909) +

infin

sum119895=ℓ

sum119896isinΛ 119895

119889(119898)

119895119896120595119895119896(119909)

119909 isin [119886 119887]

(8)

where ℓ ge 120591 and

119888(119898)

119895119896= int

119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

119889(119898)

119895119896= int

119887

119886

119891(119898)(119909) 120595

119895119896(119909) 119889119909

(9)

The second step is the estimation of 119888(119898)119895119896

and 119889(119898)119895119896

using(119884

1 119883

1) (119884

119899 119883

119899) The idea of the third step is to exploit

Advances in Statistics 3

the sparse representation of 119891(119898) by selecting the mostinteresting wavelet coefficients estimators This selection canbe of different natures (truncation thresholding ) Finallywe reconstruct these wavelet coefficients estimators on Bproviding an estimator 119891(119898) for 119891(119898)

In this study we evaluate the performance of 119891(119898) bystudying the asymptotic properties of its MISE under theassumption that 119891(119898) isin 119861119904

119901119903(119872) More precisely we aim to

determine the sharpest rate of convergence 120596119899such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862120596119899 (10)

where 119862 denotes a constant independent of 119899

3 Rates of Convergence

In this section we list the assumptions on the modelpresent our wavelet estimators and determine their rates ofconvergence under the MISE over Besov balls

31 Assumptions Let us recall that 119891 and 119892 are the functionsin (1) and ℎ is the density of119883

1

We formulate the following assumptions

(K1) We have 119891(119902)(119886) = 119891(119902)(119887) = 0 for any 119902 isin 0 119898119891(119898) isin L2([119886 119887]) and there exists a known constant1198621gt 0 such that sup

119909isin[119886119887]|119891(119909)| le 119862

1

(K2) First of all let us define the Fourier transform of anintegrable function 119906 by

F (119906) (119909) = intinfin

minusinfin

119906 (119910) 119890minus119894119909119910

119889119910 119909 isin R (11)

The notation sdotwill be used for the complex conjugateWehave119892 isin L2([1198861015840 1198871015840]) and there exist two constants1198881gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

1198881

(1 + 1199092)1205752 119909 isin R (12)

(K3) There exists a constant 1198882gt 0 such that

1198882le inf

119909isin[119886lowast 119887lowast]

ℎ (119909) (13)

The assumptions (K1) and (K3) are standard in a nonpara-metric regression framework (see for instance Tsybakov[22]) Remark that we do not need 119891(119886) = 119891(119887) = 0 forthe estimation of 119891 = 119891(0) The assumption (K2) is theso-called ldquoordinary smooth caserdquo on 119892 It is common forthe deconvolution-estimation of densities (see eg Fan andKoo [23] and Pensky and Vidakovic [24]) An example ofcompactly supported function 119892 satisfying (K2) is 119892(119909) =int2

1119910max(1 minus |119909|119910 0)119889119910 Then supp(119892) = [minus2 2] 119892 isin

L2([minus2 2]) and (K2) is satisfied with 120575 = 2 and 1198881=

min(inf119909isin[minus21205872120587]

|F(119892)(119909)| 1(41205872)) gt 0

32 When ℎ Is Known

321 Linear Wavelet Estimator We define the linear waveletestimator 119891(119898)

1by

119891(119898)

1(119909) = sum

119896isinΛ 1198950

119888(119898)

1198950 1198961206011198950 119896(119909) 119909 isin [119886 119887] (14)

where

119888(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(15)

and 1198950is an integer chosen a posteriori

Proposition 1 presents an elementary property of 119888(119898)119895119896

Proposition 1 Let 119888(119898)119895119896

be (15) and let 119888(119898)119895119896

be (9) Suppose that(K1) holds Then one has

E (119888(119898)

119895119896) = 119888

(119898)

119895119896 (16)

Theorem 2 below investigates the performance of 119891(119898)1

interms of rates of convergence under the MISE over Besovballs

Theorem 2 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119901 ge 1 119903 ge 1 119904 isin (max(1119901 minus

12 0)119873) and119873 gt 5(119898 + 120575 + 1) Let 119891(119898)1

be defined by (14)with 119895

0such that

21198950 = [119899

1(2119904lowast+2119898+2120575+1)] (17)

119904lowast= 119904 +min(12 minus 1119901 0) ([119886] denotes the integer part of 119886)Then there exists a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(18)

Note that the rate of convergence 119899minus2119904lowast(2119904lowast+2119898+2120575+1) cor-responds to the one obtained in the estimation of 119891(119898) in the1-periodic white noise convolution model with an adaptedlinear wavelet estimator (see eg Chesneau [15])

The considered estimator119891(119898)1

depends on 119904 (the smooth-ness parameter of 119891(119898)) it is not adaptiveThis aspect as wellas the rate of convergence 119899minus2119904lowast(2119904lowast+2119898+2120575+1) can be improvedwith thresholding methodsThe next paragraph is devoted toone of them the hard thresholding method

322 Hard Thresholding Wavelet Estimator Suppose that(K2) is satisfied We define the hard thresholding waveletestimator 119891(119898)

2by

119891(119898)

2(119909) = sum

119896isinΛ 120591

119888(119898)

120591119896120601120591119896(119909)

+

1198951

sum119895=120591

sum119896isinΛ 119895

119889(119898)

1198951198961|119889(119898)

119895119896|ge120581120582119895

120595119895119896(119909)

(19)

4 Advances in Statistics

119909 isin [119886 119887] where 119888(119898)119895119896

is defined by (15)

119889(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120595

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

sdot 1|(119884Vℎ(119883V))(12120587) int

infin

minusinfin(119894119909)119898((F(120595119895119896)(119909))(F(119892)(119909)))119890

minus119894119909119883V119889119909|le120589119895

(20)

1 is the indicator function 120581 gt 0 is a large enough constant1198951is the integer satisfying

21198951 = [119899

1(2119898+2120575+1)] (21)

120575 refers to (12)

120589119895= 120579

12059521198981198952120575119895radic

119899

ln 119899

120582119895= 120579

12059521198981198952120575119895radic

ln 119899119899

120579120595= (

1

120587119888211988821

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot intinfin

minusinfin

119909119898(1 + 119909

2)120575 1003816100381610038161003816F (120595) (119909)

10038161003816100381610038162

119889119909)

12

(22)

The construction of 119891(119898)2

uses the double hard threshold-ing technique introduced by Delyon and Juditsky [14] andrecently improved by Chaubey et al [25] The main interestof the thresholding using 120582

119895is to make 119891(119898)

2adaptive the

construction (and performance) of 119891(119898)2

does not dependon the knowledge of the smoothness of 119891(119898) The role ofthe thresholding using 120589

119895in (20) is to relax some usual

restrictions on themodel To bemore specific it enables us toonly suppose that 120585

1admits finite moments of order 2 (with

known E(12058521) or a known upper bound of E(1205852

1)) relaxing the

standard assumption E(|1205851|119896) lt infin for any 119896 isin N

Further details on the constructions of hard thresholdingwavelet estimators can be found in for example Donoho andJohnstone [26 27]Donoho et al [21 28]Delyon and Juditsky[14] and Hardle et al [9]

Theorem 3 below investigates the performance of 119891(119898)2

interms of rates of convergence under the MISE over Besovballs

Theorem 3 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119903 ge 1 119901 ge 2 119904 isin (0119873) or

119901 isin [1 2) 119904 isin ((2119898 + 2120575 + 1)119901119873) and119873 gt 5(119898 + 120575 + 1)Let 119891(119898)

2be defined by (19) Then there exists a constant 119862 gt 0

such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(23)

The proof of Theorem 3 is an application of a generalresult established by [25 Theorem 61] Let us mention that(ln 119899119899)2119904(2119904+2119898+2120575+1) corresponds to the rate of convergenceobtained in the estimation of 119891(119898) in the 1-periodic whitenoise convolution model with an adapted hard thresholdingwavelet estimator (see eg Chesneau [15]) In the case119898 = 0

and 120575 = 0 this rate of convergence becomes the optimalone in the minimax sense for the standard density-regressionestimation problems (see Hardle et al [9])

In comparison toTheorem 2 note that

(i) for the case 119901 ge 2 corresponding to the homogeneouszone of Besov balls (ln 119899119899)2119904(2119904+2119898+2120575+1) is equal tothe rate of convergence attained by 119891(119898)

1up to a

logarithmic term

(ii) for the case 119901 isin [1 2) corresponding to the inhomo-geneous zone of Besov balls it is significantly better interms of power

33When ℎ Is Unknown In the case where ℎ is unknown wepropose a plug-in technique which consists in estimating ℎin the construction of 119891(119898)

1(14)This yields the linear wavelet

estimator 119891(119898)3

defined by

119891(119898)

3(119909) = sum

119896isinΛ 1198952

119888(119898)

1198952 1198961206011198952 119896(119909) 119909 isin [119886 119887] (24)

where

119888(119898)

119895119896=1

119886119899

119886119899

sumV=1

119884V

ℎ (119883V)1|ℎ(119883V)|ge11988822

1

2120587

sdot intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(25)

119886119899= [1198992] 119895

2is an integer chosen a posteriori 119888

2refers to

(K3) and ℎ is an estimator of ℎ constructed from the randomvariables 119880

119899= (119883

119886119899+1 119883

119899)

There are numerous possibilities for the choice of ℎ Forinstance ℎ can be a kernel density estimator or a waveletdensity estimator (see eg Donoho et al [21] Hardle et al[9] and Juditsky and Lambert-Lacroix [29])

The estimator 119891(119898)3

is derived to the ldquoNES linear waveletestimatorrdquo introduced by Pensky and Vidakovic [16] andrecently revisited in a more simple form by Chesneau [13]

Theorem 4 below determines an upper bound of theMISE of 119891(119898)

3

Theorem 4 Suppose that (K1)ndash(K3) are satisfied ℎ isin L2([119886lowast

119887lowast]) and that 119891(119898) isin 119861119904

119901119903(119872) with 119872 gt 0 119901 ge 1 119903 ge 1

119904 isin (max(1119901 minus 12 0)119873) and 119873 gt 5(119898 + 120575 + 1) Let 119891(119898)3

Advances in Statistics 5

be defined by (24) with 1198952such that 21198952 le 119899 Then there exists

a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) 1

119899)

+ 2minus21198952119904lowast)

(26)

with 119904lowast= 119904 +min(12 minus 1119901 0)

The proof follows the idea of [13 Theorem 3] and usestechnical operations on Fourier transforms

FromTheorem 4

(i) if we chose ℎ = ℎ and 1198952= 119895

0(17) we obtain

Theorem 2(ii) if ℎ and ℎ satisfy that there exist 120592 isin [0 1] and a

constant 119862 gt 0 such that

E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) le 119862119899minus120592 (27)

then the optimal integer 1198952is such that 21198952 =

[119899120592(2119904lowast+2119898+2120575+1)] and we obtain the following rate ofconvergence for 119891(119898)

3

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast120592(2119904lowast+2119898+2120575+1)

(28)

Naturally the estimation of ℎ has a negative impact on theperformance of 119891(119898)

3 In particular if ℎ isin 119861119904

1015840

11990110158401199031015840(119872

1015840) thenthe standard density linear wavelet estimator ℎ attains therate of convergence 119899minus120592 with 120592 = 2119904

∘(2119904

∘+ 1) and 119904

∘=

1199041015840+min(12minus11199011015840 0) (and it is optimal in theminimax sensefor 1199011015840 ge 2 see Hardle et al [9]) With this choice the rate ofconvergence for 119891(119898)

3becomes 119899minus4119904lowast119904∘(2119904∘+1)(2119904lowast+2119898+2120575+1) Let

us mention that 119891(119898)3

is not adaptive since it depends on 119904However 119891(119898)

3remains an acceptable first approach for the

estimation of 119891(119898) with unknown ℎ

Conclusion and Perspectives This study considers the estima-tion of 119891(119898) from (1) According to the knowledge of ℎ ornot we propose wavelet methods and prove that they attainfast rates of convergence under the MISE over Besov ballsAmong the perspectives of this work we retain the following

(i) The relaxation of the assumption (K2) perhaps byconsidering (K21015840) there exist four constants 119862

1gt 0

120596 isin N 120578 gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816minus1

le 1198621

10038161003816100381610038161003816100381610038161003816sin(120587119909

120578)10038161003816100381610038161003816100381610038161003816

minus120596

(1 + |119909|)120575 119909 isin R (29)

This condition was first introduced by Delaigle andMeister [30] in a context of deconvolution-estimationof function It implies (K2) and has the advantage toconsider some functions 119892 having zeros in Fouriertransform domain as numerous kinds of compactlysupported functions

(ii) The construction of an adaptive version of 119891(119898)3

through the use of a thresholding method

(iii) The extension of our results to the L119901 risk with 119901 ge 1

All these aspects need further investigations that we leave forfuture works

4 Proofs

In this section 119862 denotes any constant that does not dependon 119895 119896 or 119899 Its value may change from one term to anotherand may depend on 120601 or 120595

Proof of Proposition 1 By the independence between 1198831and

1205851 E(120585

1) = 0 sup(119891 ⋆ 119892) = supp(ℎ) = [119886

lowast 119887lowast] and F(119891 ⋆

119892)(119909) = F(119891)(119909)F(119892)(119909) we have

E(1198841

ℎ (1198831)119890minus1198941199091198831) = E(

(119891 ⋆ 119892) (1198831)

ℎ (1198831)

119890minus1198941199091198831)

+ E (1205851)E(

1

ℎ (1198831)119890minus1198941199091198831)

= E((119891 ⋆ 119892) (119883

1)

ℎ (1198831)

119890minus1198941199091198831)

= int119887lowast

119886lowast

(119891 ⋆ 119892) (119910)

ℎ (119910)119890minus119894119909119910

ℎ (119910) 119889119910

= intinfin

minusinfin

(119891 ⋆ 119892) (119910) 119890minus119894119909119910

119889119910

= F (119891 ⋆ 119892) (119909)

= F (119891) (119909)F (119892) (119909)

(30)

It follows from (K1) and 119898 integration by parts that(119894119909)

119898F(119891)(119909) = F(119891(119898))(119909) Using the Fubini theorem

(119894119909)119898F(119891)(119909) = F(119891(119898))(119909) (30) and the Parseval identity

we obtain

E (119888(119898)

119895119896)

= E(1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)E(

1198841

ℎ (1198831)119890minus1198941199091198831)119889119909

6 Advances in Statistics

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)F (119891) (119909)F (119892) (119909) 119889119909

=1

2120587intinfin

minusinfin

(119894119909)119898F (119891) (119909)F (120601

119895119896) (119909) 119889119909

=1

2120587intinfin

minusinfin

F (119891(119898)) (119909)F (120601

119895119896) (119909) 119889119909

= int119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

= 119888(119898)

119895119896

(31)

Proposition 1 is proved

Proof of Theorem 2 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

0 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(32)

Using Proposition 1 (1198841 119883

1) (119884

119899 119883

119899) that are iid the

inequalitiesV(119863) le E(|119863|2) for any randomcomplex variable119863 and (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 and (K1) and (K3)we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

)

= V (119888(119898)

1198950 119896)

=1

119899V(

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

le1

(2120587)2119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le2

(2120587)2119899E(

((119891 ⋆ 119892) (1198831))2

+ 12058521

(ℎ (1198831))2

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le1

119899

2

(2120587)21198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

(33)

The Parseval identity yields

E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= int119887lowast

119886lowast

1

ℎ (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

ℎ (119910) 119889119910

le intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

F(119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

= 2120587intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

(34)

Using (K2) |F(1206011198950 119896)(119909)| = 2minus11989502|F(120601)(11990921198950)| and a change

of variables we obtain

intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

le1

11988821

intinfin

minusinfin

1199092119898(1 + 119909

2)120575 10038161003816100381610038161003816F (120601

1198950 119896) (119909)

10038161003816100381610038161003816

2

119889119909

=1

11988821

2minus1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816100381610038161003816F (120601) (

119909

21198950)1003816100381610038161003816100381610038161003816

2

119889119909

=1

11988821

intinfin

minusinfin

221198950119898119909

2119898(1 + 2

211989501199092)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le1

11988821

2(2119898+2120575)1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le 1198622(2119898+2120575)1198950

(35)

(Let us mention that intinfinminusinfin1199092119898(1 + 1199092)

120575|F(120601)(119909)|2119889119909 is finite

thanks to119873 gt 5(119898 + 120575 + 1))Combining (33) (34) and (35) we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575)1198950

1

119899 (36)

For the integer 1198950satisfying (17) it holds that

sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575+1)1198950

1

119899

le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(37)

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Statistics

integrated squared error (MISE) and the assumption that119891(119898) belongs to Besov ballsTheobtained rates of convergencecoincide with existing results for the estimation of 119891(119898) in the1-periodic convolution regression models (see for instanceChesneau [15])

The second part is devoted to the case where ℎ isunknownWe construct a new linear wavelet estimator usinga plug-in approach for the estimation of ℎ Its constructionfollows the idea of the ldquoNES linear wavelet estimatorrdquo intro-duced by Pensky and Vidakovic [16] in another regressioncontext Then we investigate its MISE properties when 119891(119898)belongs to Besov balls which naturally depend on the MISEof the considered estimator for ℎ Furthermore let us men-tion that all our results are provedwith onlymoments of order2 on 120585

1 which provides another theoretical contribution to

the subjectThe remaining part of this paper is organized as fol-

lows In Section 2 we describe some basics on wavelets andBesov balls and present our wavelet estimation methodologySection 3 is devoted to our estimators and their perfor-mances The proofs are carried out in Section 4

2 Preliminaries

This section is devoted to the presentation of the consideredwavelet basis the Besov balls and our wavelet estimationmethodology

21 Wavelet Basis Let us briefly present the wavelet basis onthe interval [119886 119887] (119886 119887) isin R2 introduced by Cohen et al [17]Let 120601 and120595 be the initial wavelet functions of the Daubechieswavelets family db2N with 119873 ge 1 (see eg Daubechies[18])These functions have the distinction of being compactlysupported and belong to the class C119886 for 119873 gt 5119886 For any119895 ge 0 and 119896 isin Z we set

120601119895119896(119909) = 2

1198952120601 (2

119895119909 minus 119896)

120595119895119896(119909) = 2

1198952120595 (2

119895119909 minus 119896)

(2)

With appropriated treatments at the boundaries thereexist an integer 120591 and a set of consecutive integers Λ

119895of

cardinality proportional to 2119895 (both depending on 119886 119887 and119873) such that for any integer ℓ ge 120591

B = 120601ℓ119896 119896 isin Λ

ℓ 120595

119895119896 119895 isin N minus 0 ℓ minus 1 119896 isin Λ

119895

(3)

forms an orthonormal basis of the space of squared integrablefunctions on [119886 119887] that is

L2([119886 119887]) = 119906 [119886 119887] 997888rarr R (int

119887

119886

|119906 (119909)|2119889119909)

12

lt infin

(4)

For the case119886 = 0 and 119887 = 1 120591 is the smallest integer satisfying2120591 ge 2119873 and Λ

119895= 0 2119895 minus 1

For any integer ℓ ge 120591 and 119906 isin L2([119886 119887]) we have thefollowing wavelet expansion

119906 (119909) = sum119896isinΛ ℓ

119888ℓ119896120601ℓ119896(119909) +

infin

sum119895=ℓ

sum119896isinΛ 119895

119889119895119896120595119895119896(119909) 119909 isin [119886 119887]

(5)

where

119888119895119896= int

119887

119886

119906 (119909) 120601119895119896(119909) 119889119909

119889119895119896= int

119887

119886

119906 (119909) 120595119895119896(119909) 119889119909

(6)

An interesting feature of the wavelet basis is to providesparse representation of 119906 only few wavelet coefficients 119889

119895119896

characterized by a high magnitude reveal the main details of119906 See for example Cohen et al [17] and Mallat [19]

22 Besov Balls We say that a function 119906 isin L2([119886 119887]) belongsto the Besov ball 119861119904

119901119903(119872) with 119904 gt 0 119901 ge 1 119903 ge 1 and119872 gt 0

if there exists a constant119862 gt 0 such that 119888119895119896

and119889119895119896

(6) satisfy

2120591(12minus1119901)

( sum119896isinΛ 120591

|119888120591119896|119901)

1119901

+ (

infin

sum119895=120591

(2119895(119904+12minus1119901)

( sum119896isinΛ 119895

10038161003816100381610038161003816119889119895119896

10038161003816100381610038161003816

119901

)

1119901

)

119903

)

1119903

le 119862

(7)

with the usual modifications if 119901 = infin or 119903 = infinThe interest of Besov balls is to contain various kinds

of homogeneous and inhomogeneous functions 119906 See forexample Meyer [20] Donoho et al [21] and Hardle et al [9]

23 Wavelet Estimation Let 119891 be the unknown function in(1) and B the considered wavelet basis taken with 119873 gt 5119898

(to ensure that 120601 and 120595 belong to the classC119898) Suppose that119891(119898) exists with 119891(119898) isin L2([119886 119887])Thefirst step in thewavelet estimation consists in expand-

ing 119891(119898) onB as

119891(119898)(119909) = sum

119896isinΛ ℓ

119888(119898)

ℓ119896120601ℓ119896(119909) +

infin

sum119895=ℓ

sum119896isinΛ 119895

119889(119898)

119895119896120595119895119896(119909)

119909 isin [119886 119887]

(8)

where ℓ ge 120591 and

119888(119898)

119895119896= int

119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

119889(119898)

119895119896= int

119887

119886

119891(119898)(119909) 120595

119895119896(119909) 119889119909

(9)

The second step is the estimation of 119888(119898)119895119896

and 119889(119898)119895119896

using(119884

1 119883

1) (119884

119899 119883

119899) The idea of the third step is to exploit

Advances in Statistics 3

the sparse representation of 119891(119898) by selecting the mostinteresting wavelet coefficients estimators This selection canbe of different natures (truncation thresholding ) Finallywe reconstruct these wavelet coefficients estimators on Bproviding an estimator 119891(119898) for 119891(119898)

In this study we evaluate the performance of 119891(119898) bystudying the asymptotic properties of its MISE under theassumption that 119891(119898) isin 119861119904

119901119903(119872) More precisely we aim to

determine the sharpest rate of convergence 120596119899such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862120596119899 (10)

where 119862 denotes a constant independent of 119899

3 Rates of Convergence

In this section we list the assumptions on the modelpresent our wavelet estimators and determine their rates ofconvergence under the MISE over Besov balls

31 Assumptions Let us recall that 119891 and 119892 are the functionsin (1) and ℎ is the density of119883

1

We formulate the following assumptions

(K1) We have 119891(119902)(119886) = 119891(119902)(119887) = 0 for any 119902 isin 0 119898119891(119898) isin L2([119886 119887]) and there exists a known constant1198621gt 0 such that sup

119909isin[119886119887]|119891(119909)| le 119862

1

(K2) First of all let us define the Fourier transform of anintegrable function 119906 by

F (119906) (119909) = intinfin

minusinfin

119906 (119910) 119890minus119894119909119910

119889119910 119909 isin R (11)

The notation sdotwill be used for the complex conjugateWehave119892 isin L2([1198861015840 1198871015840]) and there exist two constants1198881gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

1198881

(1 + 1199092)1205752 119909 isin R (12)

(K3) There exists a constant 1198882gt 0 such that

1198882le inf

119909isin[119886lowast 119887lowast]

ℎ (119909) (13)

The assumptions (K1) and (K3) are standard in a nonpara-metric regression framework (see for instance Tsybakov[22]) Remark that we do not need 119891(119886) = 119891(119887) = 0 forthe estimation of 119891 = 119891(0) The assumption (K2) is theso-called ldquoordinary smooth caserdquo on 119892 It is common forthe deconvolution-estimation of densities (see eg Fan andKoo [23] and Pensky and Vidakovic [24]) An example ofcompactly supported function 119892 satisfying (K2) is 119892(119909) =int2

1119910max(1 minus |119909|119910 0)119889119910 Then supp(119892) = [minus2 2] 119892 isin

L2([minus2 2]) and (K2) is satisfied with 120575 = 2 and 1198881=

min(inf119909isin[minus21205872120587]

|F(119892)(119909)| 1(41205872)) gt 0

32 When ℎ Is Known

321 Linear Wavelet Estimator We define the linear waveletestimator 119891(119898)

1by

119891(119898)

1(119909) = sum

119896isinΛ 1198950

119888(119898)

1198950 1198961206011198950 119896(119909) 119909 isin [119886 119887] (14)

where

119888(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(15)

and 1198950is an integer chosen a posteriori

Proposition 1 presents an elementary property of 119888(119898)119895119896

Proposition 1 Let 119888(119898)119895119896

be (15) and let 119888(119898)119895119896

be (9) Suppose that(K1) holds Then one has

E (119888(119898)

119895119896) = 119888

(119898)

119895119896 (16)

Theorem 2 below investigates the performance of 119891(119898)1

interms of rates of convergence under the MISE over Besovballs

Theorem 2 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119901 ge 1 119903 ge 1 119904 isin (max(1119901 minus

12 0)119873) and119873 gt 5(119898 + 120575 + 1) Let 119891(119898)1

be defined by (14)with 119895

0such that

21198950 = [119899

1(2119904lowast+2119898+2120575+1)] (17)

119904lowast= 119904 +min(12 minus 1119901 0) ([119886] denotes the integer part of 119886)Then there exists a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(18)

Note that the rate of convergence 119899minus2119904lowast(2119904lowast+2119898+2120575+1) cor-responds to the one obtained in the estimation of 119891(119898) in the1-periodic white noise convolution model with an adaptedlinear wavelet estimator (see eg Chesneau [15])

The considered estimator119891(119898)1

depends on 119904 (the smooth-ness parameter of 119891(119898)) it is not adaptiveThis aspect as wellas the rate of convergence 119899minus2119904lowast(2119904lowast+2119898+2120575+1) can be improvedwith thresholding methodsThe next paragraph is devoted toone of them the hard thresholding method

322 Hard Thresholding Wavelet Estimator Suppose that(K2) is satisfied We define the hard thresholding waveletestimator 119891(119898)

2by

119891(119898)

2(119909) = sum

119896isinΛ 120591

119888(119898)

120591119896120601120591119896(119909)

+

1198951

sum119895=120591

sum119896isinΛ 119895

119889(119898)

1198951198961|119889(119898)

119895119896|ge120581120582119895

120595119895119896(119909)

(19)

4 Advances in Statistics

119909 isin [119886 119887] where 119888(119898)119895119896

is defined by (15)

119889(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120595

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

sdot 1|(119884Vℎ(119883V))(12120587) int

infin

minusinfin(119894119909)119898((F(120595119895119896)(119909))(F(119892)(119909)))119890

minus119894119909119883V119889119909|le120589119895

(20)

1 is the indicator function 120581 gt 0 is a large enough constant1198951is the integer satisfying

21198951 = [119899

1(2119898+2120575+1)] (21)

120575 refers to (12)

120589119895= 120579

12059521198981198952120575119895radic

119899

ln 119899

120582119895= 120579

12059521198981198952120575119895radic

ln 119899119899

120579120595= (

1

120587119888211988821

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot intinfin

minusinfin

119909119898(1 + 119909

2)120575 1003816100381610038161003816F (120595) (119909)

10038161003816100381610038162

119889119909)

12

(22)

The construction of 119891(119898)2

uses the double hard threshold-ing technique introduced by Delyon and Juditsky [14] andrecently improved by Chaubey et al [25] The main interestof the thresholding using 120582

119895is to make 119891(119898)

2adaptive the

construction (and performance) of 119891(119898)2

does not dependon the knowledge of the smoothness of 119891(119898) The role ofthe thresholding using 120589

119895in (20) is to relax some usual

restrictions on themodel To bemore specific it enables us toonly suppose that 120585

1admits finite moments of order 2 (with

known E(12058521) or a known upper bound of E(1205852

1)) relaxing the

standard assumption E(|1205851|119896) lt infin for any 119896 isin N

Further details on the constructions of hard thresholdingwavelet estimators can be found in for example Donoho andJohnstone [26 27]Donoho et al [21 28]Delyon and Juditsky[14] and Hardle et al [9]

Theorem 3 below investigates the performance of 119891(119898)2

interms of rates of convergence under the MISE over Besovballs

Theorem 3 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119903 ge 1 119901 ge 2 119904 isin (0119873) or

119901 isin [1 2) 119904 isin ((2119898 + 2120575 + 1)119901119873) and119873 gt 5(119898 + 120575 + 1)Let 119891(119898)

2be defined by (19) Then there exists a constant 119862 gt 0

such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(23)

The proof of Theorem 3 is an application of a generalresult established by [25 Theorem 61] Let us mention that(ln 119899119899)2119904(2119904+2119898+2120575+1) corresponds to the rate of convergenceobtained in the estimation of 119891(119898) in the 1-periodic whitenoise convolution model with an adapted hard thresholdingwavelet estimator (see eg Chesneau [15]) In the case119898 = 0

and 120575 = 0 this rate of convergence becomes the optimalone in the minimax sense for the standard density-regressionestimation problems (see Hardle et al [9])

In comparison toTheorem 2 note that

(i) for the case 119901 ge 2 corresponding to the homogeneouszone of Besov balls (ln 119899119899)2119904(2119904+2119898+2120575+1) is equal tothe rate of convergence attained by 119891(119898)

1up to a

logarithmic term

(ii) for the case 119901 isin [1 2) corresponding to the inhomo-geneous zone of Besov balls it is significantly better interms of power

33When ℎ Is Unknown In the case where ℎ is unknown wepropose a plug-in technique which consists in estimating ℎin the construction of 119891(119898)

1(14)This yields the linear wavelet

estimator 119891(119898)3

defined by

119891(119898)

3(119909) = sum

119896isinΛ 1198952

119888(119898)

1198952 1198961206011198952 119896(119909) 119909 isin [119886 119887] (24)

where

119888(119898)

119895119896=1

119886119899

119886119899

sumV=1

119884V

ℎ (119883V)1|ℎ(119883V)|ge11988822

1

2120587

sdot intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(25)

119886119899= [1198992] 119895

2is an integer chosen a posteriori 119888

2refers to

(K3) and ℎ is an estimator of ℎ constructed from the randomvariables 119880

119899= (119883

119886119899+1 119883

119899)

There are numerous possibilities for the choice of ℎ Forinstance ℎ can be a kernel density estimator or a waveletdensity estimator (see eg Donoho et al [21] Hardle et al[9] and Juditsky and Lambert-Lacroix [29])

The estimator 119891(119898)3

is derived to the ldquoNES linear waveletestimatorrdquo introduced by Pensky and Vidakovic [16] andrecently revisited in a more simple form by Chesneau [13]

Theorem 4 below determines an upper bound of theMISE of 119891(119898)

3

Theorem 4 Suppose that (K1)ndash(K3) are satisfied ℎ isin L2([119886lowast

119887lowast]) and that 119891(119898) isin 119861119904

119901119903(119872) with 119872 gt 0 119901 ge 1 119903 ge 1

119904 isin (max(1119901 minus 12 0)119873) and 119873 gt 5(119898 + 120575 + 1) Let 119891(119898)3

Advances in Statistics 5

be defined by (24) with 1198952such that 21198952 le 119899 Then there exists

a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) 1

119899)

+ 2minus21198952119904lowast)

(26)

with 119904lowast= 119904 +min(12 minus 1119901 0)

The proof follows the idea of [13 Theorem 3] and usestechnical operations on Fourier transforms

FromTheorem 4

(i) if we chose ℎ = ℎ and 1198952= 119895

0(17) we obtain

Theorem 2(ii) if ℎ and ℎ satisfy that there exist 120592 isin [0 1] and a

constant 119862 gt 0 such that

E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) le 119862119899minus120592 (27)

then the optimal integer 1198952is such that 21198952 =

[119899120592(2119904lowast+2119898+2120575+1)] and we obtain the following rate ofconvergence for 119891(119898)

3

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast120592(2119904lowast+2119898+2120575+1)

(28)

Naturally the estimation of ℎ has a negative impact on theperformance of 119891(119898)

3 In particular if ℎ isin 119861119904

1015840

11990110158401199031015840(119872

1015840) thenthe standard density linear wavelet estimator ℎ attains therate of convergence 119899minus120592 with 120592 = 2119904

∘(2119904

∘+ 1) and 119904

∘=

1199041015840+min(12minus11199011015840 0) (and it is optimal in theminimax sensefor 1199011015840 ge 2 see Hardle et al [9]) With this choice the rate ofconvergence for 119891(119898)

3becomes 119899minus4119904lowast119904∘(2119904∘+1)(2119904lowast+2119898+2120575+1) Let

us mention that 119891(119898)3

is not adaptive since it depends on 119904However 119891(119898)

3remains an acceptable first approach for the

estimation of 119891(119898) with unknown ℎ

Conclusion and Perspectives This study considers the estima-tion of 119891(119898) from (1) According to the knowledge of ℎ ornot we propose wavelet methods and prove that they attainfast rates of convergence under the MISE over Besov ballsAmong the perspectives of this work we retain the following

(i) The relaxation of the assumption (K2) perhaps byconsidering (K21015840) there exist four constants 119862

1gt 0

120596 isin N 120578 gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816minus1

le 1198621

10038161003816100381610038161003816100381610038161003816sin(120587119909

120578)10038161003816100381610038161003816100381610038161003816

minus120596

(1 + |119909|)120575 119909 isin R (29)

This condition was first introduced by Delaigle andMeister [30] in a context of deconvolution-estimationof function It implies (K2) and has the advantage toconsider some functions 119892 having zeros in Fouriertransform domain as numerous kinds of compactlysupported functions

(ii) The construction of an adaptive version of 119891(119898)3

through the use of a thresholding method

(iii) The extension of our results to the L119901 risk with 119901 ge 1

All these aspects need further investigations that we leave forfuture works

4 Proofs

In this section 119862 denotes any constant that does not dependon 119895 119896 or 119899 Its value may change from one term to anotherand may depend on 120601 or 120595

Proof of Proposition 1 By the independence between 1198831and

1205851 E(120585

1) = 0 sup(119891 ⋆ 119892) = supp(ℎ) = [119886

lowast 119887lowast] and F(119891 ⋆

119892)(119909) = F(119891)(119909)F(119892)(119909) we have

E(1198841

ℎ (1198831)119890minus1198941199091198831) = E(

(119891 ⋆ 119892) (1198831)

ℎ (1198831)

119890minus1198941199091198831)

+ E (1205851)E(

1

ℎ (1198831)119890minus1198941199091198831)

= E((119891 ⋆ 119892) (119883

1)

ℎ (1198831)

119890minus1198941199091198831)

= int119887lowast

119886lowast

(119891 ⋆ 119892) (119910)

ℎ (119910)119890minus119894119909119910

ℎ (119910) 119889119910

= intinfin

minusinfin

(119891 ⋆ 119892) (119910) 119890minus119894119909119910

119889119910

= F (119891 ⋆ 119892) (119909)

= F (119891) (119909)F (119892) (119909)

(30)

It follows from (K1) and 119898 integration by parts that(119894119909)

119898F(119891)(119909) = F(119891(119898))(119909) Using the Fubini theorem

(119894119909)119898F(119891)(119909) = F(119891(119898))(119909) (30) and the Parseval identity

we obtain

E (119888(119898)

119895119896)

= E(1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)E(

1198841

ℎ (1198831)119890minus1198941199091198831)119889119909

6 Advances in Statistics

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)F (119891) (119909)F (119892) (119909) 119889119909

=1

2120587intinfin

minusinfin

(119894119909)119898F (119891) (119909)F (120601

119895119896) (119909) 119889119909

=1

2120587intinfin

minusinfin

F (119891(119898)) (119909)F (120601

119895119896) (119909) 119889119909

= int119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

= 119888(119898)

119895119896

(31)

Proposition 1 is proved

Proof of Theorem 2 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

0 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(32)

Using Proposition 1 (1198841 119883

1) (119884

119899 119883

119899) that are iid the

inequalitiesV(119863) le E(|119863|2) for any randomcomplex variable119863 and (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 and (K1) and (K3)we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

)

= V (119888(119898)

1198950 119896)

=1

119899V(

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

le1

(2120587)2119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le2

(2120587)2119899E(

((119891 ⋆ 119892) (1198831))2

+ 12058521

(ℎ (1198831))2

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le1

119899

2

(2120587)21198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

(33)

The Parseval identity yields

E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= int119887lowast

119886lowast

1

ℎ (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

ℎ (119910) 119889119910

le intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

F(119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

= 2120587intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

(34)

Using (K2) |F(1206011198950 119896)(119909)| = 2minus11989502|F(120601)(11990921198950)| and a change

of variables we obtain

intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

le1

11988821

intinfin

minusinfin

1199092119898(1 + 119909

2)120575 10038161003816100381610038161003816F (120601

1198950 119896) (119909)

10038161003816100381610038161003816

2

119889119909

=1

11988821

2minus1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816100381610038161003816F (120601) (

119909

21198950)1003816100381610038161003816100381610038161003816

2

119889119909

=1

11988821

intinfin

minusinfin

221198950119898119909

2119898(1 + 2

211989501199092)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le1

11988821

2(2119898+2120575)1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le 1198622(2119898+2120575)1198950

(35)

(Let us mention that intinfinminusinfin1199092119898(1 + 1199092)

120575|F(120601)(119909)|2119889119909 is finite

thanks to119873 gt 5(119898 + 120575 + 1))Combining (33) (34) and (35) we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575)1198950

1

119899 (36)

For the integer 1198950satisfying (17) it holds that

sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575+1)1198950

1

119899

le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(37)

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 3

the sparse representation of 119891(119898) by selecting the mostinteresting wavelet coefficients estimators This selection canbe of different natures (truncation thresholding ) Finallywe reconstruct these wavelet coefficients estimators on Bproviding an estimator 119891(119898) for 119891(119898)

In this study we evaluate the performance of 119891(119898) bystudying the asymptotic properties of its MISE under theassumption that 119891(119898) isin 119861119904

119901119903(119872) More precisely we aim to

determine the sharpest rate of convergence 120596119899such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862120596119899 (10)

where 119862 denotes a constant independent of 119899

3 Rates of Convergence

In this section we list the assumptions on the modelpresent our wavelet estimators and determine their rates ofconvergence under the MISE over Besov balls

31 Assumptions Let us recall that 119891 and 119892 are the functionsin (1) and ℎ is the density of119883

1

We formulate the following assumptions

(K1) We have 119891(119902)(119886) = 119891(119902)(119887) = 0 for any 119902 isin 0 119898119891(119898) isin L2([119886 119887]) and there exists a known constant1198621gt 0 such that sup

119909isin[119886119887]|119891(119909)| le 119862

1

(K2) First of all let us define the Fourier transform of anintegrable function 119906 by

F (119906) (119909) = intinfin

minusinfin

119906 (119910) 119890minus119894119909119910

119889119910 119909 isin R (11)

The notation sdotwill be used for the complex conjugateWehave119892 isin L2([1198861015840 1198871015840]) and there exist two constants1198881gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816 ge

1198881

(1 + 1199092)1205752 119909 isin R (12)

(K3) There exists a constant 1198882gt 0 such that

1198882le inf

119909isin[119886lowast 119887lowast]

ℎ (119909) (13)

The assumptions (K1) and (K3) are standard in a nonpara-metric regression framework (see for instance Tsybakov[22]) Remark that we do not need 119891(119886) = 119891(119887) = 0 forthe estimation of 119891 = 119891(0) The assumption (K2) is theso-called ldquoordinary smooth caserdquo on 119892 It is common forthe deconvolution-estimation of densities (see eg Fan andKoo [23] and Pensky and Vidakovic [24]) An example ofcompactly supported function 119892 satisfying (K2) is 119892(119909) =int2

1119910max(1 minus |119909|119910 0)119889119910 Then supp(119892) = [minus2 2] 119892 isin

L2([minus2 2]) and (K2) is satisfied with 120575 = 2 and 1198881=

min(inf119909isin[minus21205872120587]

|F(119892)(119909)| 1(41205872)) gt 0

32 When ℎ Is Known

321 Linear Wavelet Estimator We define the linear waveletestimator 119891(119898)

1by

119891(119898)

1(119909) = sum

119896isinΛ 1198950

119888(119898)

1198950 1198961206011198950 119896(119909) 119909 isin [119886 119887] (14)

where

119888(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(15)

and 1198950is an integer chosen a posteriori

Proposition 1 presents an elementary property of 119888(119898)119895119896

Proposition 1 Let 119888(119898)119895119896

be (15) and let 119888(119898)119895119896

be (9) Suppose that(K1) holds Then one has

E (119888(119898)

119895119896) = 119888

(119898)

119895119896 (16)

Theorem 2 below investigates the performance of 119891(119898)1

interms of rates of convergence under the MISE over Besovballs

Theorem 2 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119901 ge 1 119903 ge 1 119904 isin (max(1119901 minus

12 0)119873) and119873 gt 5(119898 + 120575 + 1) Let 119891(119898)1

be defined by (14)with 119895

0such that

21198950 = [119899

1(2119904lowast+2119898+2120575+1)] (17)

119904lowast= 119904 +min(12 minus 1119901 0) ([119886] denotes the integer part of 119886)Then there exists a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(18)

Note that the rate of convergence 119899minus2119904lowast(2119904lowast+2119898+2120575+1) cor-responds to the one obtained in the estimation of 119891(119898) in the1-periodic white noise convolution model with an adaptedlinear wavelet estimator (see eg Chesneau [15])

The considered estimator119891(119898)1

depends on 119904 (the smooth-ness parameter of 119891(119898)) it is not adaptiveThis aspect as wellas the rate of convergence 119899minus2119904lowast(2119904lowast+2119898+2120575+1) can be improvedwith thresholding methodsThe next paragraph is devoted toone of them the hard thresholding method

322 Hard Thresholding Wavelet Estimator Suppose that(K2) is satisfied We define the hard thresholding waveletestimator 119891(119898)

2by

119891(119898)

2(119909) = sum

119896isinΛ 120591

119888(119898)

120591119896120601120591119896(119909)

+

1198951

sum119895=120591

sum119896isinΛ 119895

119889(119898)

1198951198961|119889(119898)

119895119896|ge120581120582119895

120595119895119896(119909)

(19)

4 Advances in Statistics

119909 isin [119886 119887] where 119888(119898)119895119896

is defined by (15)

119889(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120595

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

sdot 1|(119884Vℎ(119883V))(12120587) int

infin

minusinfin(119894119909)119898((F(120595119895119896)(119909))(F(119892)(119909)))119890

minus119894119909119883V119889119909|le120589119895

(20)

1 is the indicator function 120581 gt 0 is a large enough constant1198951is the integer satisfying

21198951 = [119899

1(2119898+2120575+1)] (21)

120575 refers to (12)

120589119895= 120579

12059521198981198952120575119895radic

119899

ln 119899

120582119895= 120579

12059521198981198952120575119895radic

ln 119899119899

120579120595= (

1

120587119888211988821

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot intinfin

minusinfin

119909119898(1 + 119909

2)120575 1003816100381610038161003816F (120595) (119909)

10038161003816100381610038162

119889119909)

12

(22)

The construction of 119891(119898)2

uses the double hard threshold-ing technique introduced by Delyon and Juditsky [14] andrecently improved by Chaubey et al [25] The main interestof the thresholding using 120582

119895is to make 119891(119898)

2adaptive the

construction (and performance) of 119891(119898)2

does not dependon the knowledge of the smoothness of 119891(119898) The role ofthe thresholding using 120589

119895in (20) is to relax some usual

restrictions on themodel To bemore specific it enables us toonly suppose that 120585

1admits finite moments of order 2 (with

known E(12058521) or a known upper bound of E(1205852

1)) relaxing the

standard assumption E(|1205851|119896) lt infin for any 119896 isin N

Further details on the constructions of hard thresholdingwavelet estimators can be found in for example Donoho andJohnstone [26 27]Donoho et al [21 28]Delyon and Juditsky[14] and Hardle et al [9]

Theorem 3 below investigates the performance of 119891(119898)2

interms of rates of convergence under the MISE over Besovballs

Theorem 3 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119903 ge 1 119901 ge 2 119904 isin (0119873) or

119901 isin [1 2) 119904 isin ((2119898 + 2120575 + 1)119901119873) and119873 gt 5(119898 + 120575 + 1)Let 119891(119898)

2be defined by (19) Then there exists a constant 119862 gt 0

such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(23)

The proof of Theorem 3 is an application of a generalresult established by [25 Theorem 61] Let us mention that(ln 119899119899)2119904(2119904+2119898+2120575+1) corresponds to the rate of convergenceobtained in the estimation of 119891(119898) in the 1-periodic whitenoise convolution model with an adapted hard thresholdingwavelet estimator (see eg Chesneau [15]) In the case119898 = 0

and 120575 = 0 this rate of convergence becomes the optimalone in the minimax sense for the standard density-regressionestimation problems (see Hardle et al [9])

In comparison toTheorem 2 note that

(i) for the case 119901 ge 2 corresponding to the homogeneouszone of Besov balls (ln 119899119899)2119904(2119904+2119898+2120575+1) is equal tothe rate of convergence attained by 119891(119898)

1up to a

logarithmic term

(ii) for the case 119901 isin [1 2) corresponding to the inhomo-geneous zone of Besov balls it is significantly better interms of power

33When ℎ Is Unknown In the case where ℎ is unknown wepropose a plug-in technique which consists in estimating ℎin the construction of 119891(119898)

1(14)This yields the linear wavelet

estimator 119891(119898)3

defined by

119891(119898)

3(119909) = sum

119896isinΛ 1198952

119888(119898)

1198952 1198961206011198952 119896(119909) 119909 isin [119886 119887] (24)

where

119888(119898)

119895119896=1

119886119899

119886119899

sumV=1

119884V

ℎ (119883V)1|ℎ(119883V)|ge11988822

1

2120587

sdot intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(25)

119886119899= [1198992] 119895

2is an integer chosen a posteriori 119888

2refers to

(K3) and ℎ is an estimator of ℎ constructed from the randomvariables 119880

119899= (119883

119886119899+1 119883

119899)

There are numerous possibilities for the choice of ℎ Forinstance ℎ can be a kernel density estimator or a waveletdensity estimator (see eg Donoho et al [21] Hardle et al[9] and Juditsky and Lambert-Lacroix [29])

The estimator 119891(119898)3

is derived to the ldquoNES linear waveletestimatorrdquo introduced by Pensky and Vidakovic [16] andrecently revisited in a more simple form by Chesneau [13]

Theorem 4 below determines an upper bound of theMISE of 119891(119898)

3

Theorem 4 Suppose that (K1)ndash(K3) are satisfied ℎ isin L2([119886lowast

119887lowast]) and that 119891(119898) isin 119861119904

119901119903(119872) with 119872 gt 0 119901 ge 1 119903 ge 1

119904 isin (max(1119901 minus 12 0)119873) and 119873 gt 5(119898 + 120575 + 1) Let 119891(119898)3

Advances in Statistics 5

be defined by (24) with 1198952such that 21198952 le 119899 Then there exists

a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) 1

119899)

+ 2minus21198952119904lowast)

(26)

with 119904lowast= 119904 +min(12 minus 1119901 0)

The proof follows the idea of [13 Theorem 3] and usestechnical operations on Fourier transforms

FromTheorem 4

(i) if we chose ℎ = ℎ and 1198952= 119895

0(17) we obtain

Theorem 2(ii) if ℎ and ℎ satisfy that there exist 120592 isin [0 1] and a

constant 119862 gt 0 such that

E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) le 119862119899minus120592 (27)

then the optimal integer 1198952is such that 21198952 =

[119899120592(2119904lowast+2119898+2120575+1)] and we obtain the following rate ofconvergence for 119891(119898)

3

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast120592(2119904lowast+2119898+2120575+1)

(28)

Naturally the estimation of ℎ has a negative impact on theperformance of 119891(119898)

3 In particular if ℎ isin 119861119904

1015840

11990110158401199031015840(119872

1015840) thenthe standard density linear wavelet estimator ℎ attains therate of convergence 119899minus120592 with 120592 = 2119904

∘(2119904

∘+ 1) and 119904

∘=

1199041015840+min(12minus11199011015840 0) (and it is optimal in theminimax sensefor 1199011015840 ge 2 see Hardle et al [9]) With this choice the rate ofconvergence for 119891(119898)

3becomes 119899minus4119904lowast119904∘(2119904∘+1)(2119904lowast+2119898+2120575+1) Let

us mention that 119891(119898)3

is not adaptive since it depends on 119904However 119891(119898)

3remains an acceptable first approach for the

estimation of 119891(119898) with unknown ℎ

Conclusion and Perspectives This study considers the estima-tion of 119891(119898) from (1) According to the knowledge of ℎ ornot we propose wavelet methods and prove that they attainfast rates of convergence under the MISE over Besov ballsAmong the perspectives of this work we retain the following

(i) The relaxation of the assumption (K2) perhaps byconsidering (K21015840) there exist four constants 119862

1gt 0

120596 isin N 120578 gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816minus1

le 1198621

10038161003816100381610038161003816100381610038161003816sin(120587119909

120578)10038161003816100381610038161003816100381610038161003816

minus120596

(1 + |119909|)120575 119909 isin R (29)

This condition was first introduced by Delaigle andMeister [30] in a context of deconvolution-estimationof function It implies (K2) and has the advantage toconsider some functions 119892 having zeros in Fouriertransform domain as numerous kinds of compactlysupported functions

(ii) The construction of an adaptive version of 119891(119898)3

through the use of a thresholding method

(iii) The extension of our results to the L119901 risk with 119901 ge 1

All these aspects need further investigations that we leave forfuture works

4 Proofs

In this section 119862 denotes any constant that does not dependon 119895 119896 or 119899 Its value may change from one term to anotherand may depend on 120601 or 120595

Proof of Proposition 1 By the independence between 1198831and

1205851 E(120585

1) = 0 sup(119891 ⋆ 119892) = supp(ℎ) = [119886

lowast 119887lowast] and F(119891 ⋆

119892)(119909) = F(119891)(119909)F(119892)(119909) we have

E(1198841

ℎ (1198831)119890minus1198941199091198831) = E(

(119891 ⋆ 119892) (1198831)

ℎ (1198831)

119890minus1198941199091198831)

+ E (1205851)E(

1

ℎ (1198831)119890minus1198941199091198831)

= E((119891 ⋆ 119892) (119883

1)

ℎ (1198831)

119890minus1198941199091198831)

= int119887lowast

119886lowast

(119891 ⋆ 119892) (119910)

ℎ (119910)119890minus119894119909119910

ℎ (119910) 119889119910

= intinfin

minusinfin

(119891 ⋆ 119892) (119910) 119890minus119894119909119910

119889119910

= F (119891 ⋆ 119892) (119909)

= F (119891) (119909)F (119892) (119909)

(30)

It follows from (K1) and 119898 integration by parts that(119894119909)

119898F(119891)(119909) = F(119891(119898))(119909) Using the Fubini theorem

(119894119909)119898F(119891)(119909) = F(119891(119898))(119909) (30) and the Parseval identity

we obtain

E (119888(119898)

119895119896)

= E(1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)E(

1198841

ℎ (1198831)119890minus1198941199091198831)119889119909

6 Advances in Statistics

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)F (119891) (119909)F (119892) (119909) 119889119909

=1

2120587intinfin

minusinfin

(119894119909)119898F (119891) (119909)F (120601

119895119896) (119909) 119889119909

=1

2120587intinfin

minusinfin

F (119891(119898)) (119909)F (120601

119895119896) (119909) 119889119909

= int119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

= 119888(119898)

119895119896

(31)

Proposition 1 is proved

Proof of Theorem 2 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

0 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(32)

Using Proposition 1 (1198841 119883

1) (119884

119899 119883

119899) that are iid the

inequalitiesV(119863) le E(|119863|2) for any randomcomplex variable119863 and (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 and (K1) and (K3)we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

)

= V (119888(119898)

1198950 119896)

=1

119899V(

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

le1

(2120587)2119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le2

(2120587)2119899E(

((119891 ⋆ 119892) (1198831))2

+ 12058521

(ℎ (1198831))2

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le1

119899

2

(2120587)21198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

(33)

The Parseval identity yields

E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= int119887lowast

119886lowast

1

ℎ (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

ℎ (119910) 119889119910

le intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

F(119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

= 2120587intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

(34)

Using (K2) |F(1206011198950 119896)(119909)| = 2minus11989502|F(120601)(11990921198950)| and a change

of variables we obtain

intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

le1

11988821

intinfin

minusinfin

1199092119898(1 + 119909

2)120575 10038161003816100381610038161003816F (120601

1198950 119896) (119909)

10038161003816100381610038161003816

2

119889119909

=1

11988821

2minus1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816100381610038161003816F (120601) (

119909

21198950)1003816100381610038161003816100381610038161003816

2

119889119909

=1

11988821

intinfin

minusinfin

221198950119898119909

2119898(1 + 2

211989501199092)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le1

11988821

2(2119898+2120575)1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le 1198622(2119898+2120575)1198950

(35)

(Let us mention that intinfinminusinfin1199092119898(1 + 1199092)

120575|F(120601)(119909)|2119889119909 is finite

thanks to119873 gt 5(119898 + 120575 + 1))Combining (33) (34) and (35) we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575)1198950

1

119899 (36)

For the integer 1198950satisfying (17) it holds that

sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575+1)1198950

1

119899

le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(37)

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Statistics

119909 isin [119886 119887] where 119888(119898)119895119896

is defined by (15)

119889(119898)

119895119896=1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120595

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

sdot 1|(119884Vℎ(119883V))(12120587) int

infin

minusinfin(119894119909)119898((F(120595119895119896)(119909))(F(119892)(119909)))119890

minus119894119909119883V119889119909|le120589119895

(20)

1 is the indicator function 120581 gt 0 is a large enough constant1198951is the integer satisfying

21198951 = [119899

1(2119898+2120575+1)] (21)

120575 refers to (12)

120589119895= 120579

12059521198981198952120575119895radic

119899

ln 119899

120582119895= 120579

12059521198981198952120575119895radic

ln 119899119899

120579120595= (

1

120587119888211988821

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot intinfin

minusinfin

119909119898(1 + 119909

2)120575 1003816100381610038161003816F (120595) (119909)

10038161003816100381610038162

119889119909)

12

(22)

The construction of 119891(119898)2

uses the double hard threshold-ing technique introduced by Delyon and Juditsky [14] andrecently improved by Chaubey et al [25] The main interestof the thresholding using 120582

119895is to make 119891(119898)

2adaptive the

construction (and performance) of 119891(119898)2

does not dependon the knowledge of the smoothness of 119891(119898) The role ofthe thresholding using 120589

119895in (20) is to relax some usual

restrictions on themodel To bemore specific it enables us toonly suppose that 120585

1admits finite moments of order 2 (with

known E(12058521) or a known upper bound of E(1205852

1)) relaxing the

standard assumption E(|1205851|119896) lt infin for any 119896 isin N

Further details on the constructions of hard thresholdingwavelet estimators can be found in for example Donoho andJohnstone [26 27]Donoho et al [21 28]Delyon and Juditsky[14] and Hardle et al [9]

Theorem 3 below investigates the performance of 119891(119898)2

interms of rates of convergence under the MISE over Besovballs

Theorem 3 Suppose that (K1)ndash(K3) are satisfied and that119891(119898) isin 119861119904

119901119903(119872) with119872 gt 0 119903 ge 1 119901 ge 2 119904 isin (0119873) or

119901 isin [1 2) 119904 isin ((2119898 + 2120575 + 1)119901119873) and119873 gt 5(119898 + 120575 + 1)Let 119891(119898)

2be defined by (19) Then there exists a constant 119862 gt 0

such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(23)

The proof of Theorem 3 is an application of a generalresult established by [25 Theorem 61] Let us mention that(ln 119899119899)2119904(2119904+2119898+2120575+1) corresponds to the rate of convergenceobtained in the estimation of 119891(119898) in the 1-periodic whitenoise convolution model with an adapted hard thresholdingwavelet estimator (see eg Chesneau [15]) In the case119898 = 0

and 120575 = 0 this rate of convergence becomes the optimalone in the minimax sense for the standard density-regressionestimation problems (see Hardle et al [9])

In comparison toTheorem 2 note that

(i) for the case 119901 ge 2 corresponding to the homogeneouszone of Besov balls (ln 119899119899)2119904(2119904+2119898+2120575+1) is equal tothe rate of convergence attained by 119891(119898)

1up to a

logarithmic term

(ii) for the case 119901 isin [1 2) corresponding to the inhomo-geneous zone of Besov balls it is significantly better interms of power

33When ℎ Is Unknown In the case where ℎ is unknown wepropose a plug-in technique which consists in estimating ℎin the construction of 119891(119898)

1(14)This yields the linear wavelet

estimator 119891(119898)3

defined by

119891(119898)

3(119909) = sum

119896isinΛ 1198952

119888(119898)

1198952 1198961206011198952 119896(119909) 119909 isin [119886 119887] (24)

where

119888(119898)

119895119896=1

119886119899

119886119899

sumV=1

119884V

ℎ (119883V)1|ℎ(119883V)|ge11988822

1

2120587

sdot intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

(25)

119886119899= [1198992] 119895

2is an integer chosen a posteriori 119888

2refers to

(K3) and ℎ is an estimator of ℎ constructed from the randomvariables 119880

119899= (119883

119886119899+1 119883

119899)

There are numerous possibilities for the choice of ℎ Forinstance ℎ can be a kernel density estimator or a waveletdensity estimator (see eg Donoho et al [21] Hardle et al[9] and Juditsky and Lambert-Lacroix [29])

The estimator 119891(119898)3

is derived to the ldquoNES linear waveletestimatorrdquo introduced by Pensky and Vidakovic [16] andrecently revisited in a more simple form by Chesneau [13]

Theorem 4 below determines an upper bound of theMISE of 119891(119898)

3

Theorem 4 Suppose that (K1)ndash(K3) are satisfied ℎ isin L2([119886lowast

119887lowast]) and that 119891(119898) isin 119861119904

119901119903(119872) with 119872 gt 0 119901 ge 1 119903 ge 1

119904 isin (max(1119901 minus 12 0)119873) and 119873 gt 5(119898 + 120575 + 1) Let 119891(119898)3

Advances in Statistics 5

be defined by (24) with 1198952such that 21198952 le 119899 Then there exists

a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) 1

119899)

+ 2minus21198952119904lowast)

(26)

with 119904lowast= 119904 +min(12 minus 1119901 0)

The proof follows the idea of [13 Theorem 3] and usestechnical operations on Fourier transforms

FromTheorem 4

(i) if we chose ℎ = ℎ and 1198952= 119895

0(17) we obtain

Theorem 2(ii) if ℎ and ℎ satisfy that there exist 120592 isin [0 1] and a

constant 119862 gt 0 such that

E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) le 119862119899minus120592 (27)

then the optimal integer 1198952is such that 21198952 =

[119899120592(2119904lowast+2119898+2120575+1)] and we obtain the following rate ofconvergence for 119891(119898)

3

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast120592(2119904lowast+2119898+2120575+1)

(28)

Naturally the estimation of ℎ has a negative impact on theperformance of 119891(119898)

3 In particular if ℎ isin 119861119904

1015840

11990110158401199031015840(119872

1015840) thenthe standard density linear wavelet estimator ℎ attains therate of convergence 119899minus120592 with 120592 = 2119904

∘(2119904

∘+ 1) and 119904

∘=

1199041015840+min(12minus11199011015840 0) (and it is optimal in theminimax sensefor 1199011015840 ge 2 see Hardle et al [9]) With this choice the rate ofconvergence for 119891(119898)

3becomes 119899minus4119904lowast119904∘(2119904∘+1)(2119904lowast+2119898+2120575+1) Let

us mention that 119891(119898)3

is not adaptive since it depends on 119904However 119891(119898)

3remains an acceptable first approach for the

estimation of 119891(119898) with unknown ℎ

Conclusion and Perspectives This study considers the estima-tion of 119891(119898) from (1) According to the knowledge of ℎ ornot we propose wavelet methods and prove that they attainfast rates of convergence under the MISE over Besov ballsAmong the perspectives of this work we retain the following

(i) The relaxation of the assumption (K2) perhaps byconsidering (K21015840) there exist four constants 119862

1gt 0

120596 isin N 120578 gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816minus1

le 1198621

10038161003816100381610038161003816100381610038161003816sin(120587119909

120578)10038161003816100381610038161003816100381610038161003816

minus120596

(1 + |119909|)120575 119909 isin R (29)

This condition was first introduced by Delaigle andMeister [30] in a context of deconvolution-estimationof function It implies (K2) and has the advantage toconsider some functions 119892 having zeros in Fouriertransform domain as numerous kinds of compactlysupported functions

(ii) The construction of an adaptive version of 119891(119898)3

through the use of a thresholding method

(iii) The extension of our results to the L119901 risk with 119901 ge 1

All these aspects need further investigations that we leave forfuture works

4 Proofs

In this section 119862 denotes any constant that does not dependon 119895 119896 or 119899 Its value may change from one term to anotherand may depend on 120601 or 120595

Proof of Proposition 1 By the independence between 1198831and

1205851 E(120585

1) = 0 sup(119891 ⋆ 119892) = supp(ℎ) = [119886

lowast 119887lowast] and F(119891 ⋆

119892)(119909) = F(119891)(119909)F(119892)(119909) we have

E(1198841

ℎ (1198831)119890minus1198941199091198831) = E(

(119891 ⋆ 119892) (1198831)

ℎ (1198831)

119890minus1198941199091198831)

+ E (1205851)E(

1

ℎ (1198831)119890minus1198941199091198831)

= E((119891 ⋆ 119892) (119883

1)

ℎ (1198831)

119890minus1198941199091198831)

= int119887lowast

119886lowast

(119891 ⋆ 119892) (119910)

ℎ (119910)119890minus119894119909119910

ℎ (119910) 119889119910

= intinfin

minusinfin

(119891 ⋆ 119892) (119910) 119890minus119894119909119910

119889119910

= F (119891 ⋆ 119892) (119909)

= F (119891) (119909)F (119892) (119909)

(30)

It follows from (K1) and 119898 integration by parts that(119894119909)

119898F(119891)(119909) = F(119891(119898))(119909) Using the Fubini theorem

(119894119909)119898F(119891)(119909) = F(119891(119898))(119909) (30) and the Parseval identity

we obtain

E (119888(119898)

119895119896)

= E(1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)E(

1198841

ℎ (1198831)119890minus1198941199091198831)119889119909

6 Advances in Statistics

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)F (119891) (119909)F (119892) (119909) 119889119909

=1

2120587intinfin

minusinfin

(119894119909)119898F (119891) (119909)F (120601

119895119896) (119909) 119889119909

=1

2120587intinfin

minusinfin

F (119891(119898)) (119909)F (120601

119895119896) (119909) 119889119909

= int119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

= 119888(119898)

119895119896

(31)

Proposition 1 is proved

Proof of Theorem 2 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

0 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(32)

Using Proposition 1 (1198841 119883

1) (119884

119899 119883

119899) that are iid the

inequalitiesV(119863) le E(|119863|2) for any randomcomplex variable119863 and (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 and (K1) and (K3)we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

)

= V (119888(119898)

1198950 119896)

=1

119899V(

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

le1

(2120587)2119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le2

(2120587)2119899E(

((119891 ⋆ 119892) (1198831))2

+ 12058521

(ℎ (1198831))2

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le1

119899

2

(2120587)21198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

(33)

The Parseval identity yields

E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= int119887lowast

119886lowast

1

ℎ (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

ℎ (119910) 119889119910

le intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

F(119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

= 2120587intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

(34)

Using (K2) |F(1206011198950 119896)(119909)| = 2minus11989502|F(120601)(11990921198950)| and a change

of variables we obtain

intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

le1

11988821

intinfin

minusinfin

1199092119898(1 + 119909

2)120575 10038161003816100381610038161003816F (120601

1198950 119896) (119909)

10038161003816100381610038161003816

2

119889119909

=1

11988821

2minus1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816100381610038161003816F (120601) (

119909

21198950)1003816100381610038161003816100381610038161003816

2

119889119909

=1

11988821

intinfin

minusinfin

221198950119898119909

2119898(1 + 2

211989501199092)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le1

11988821

2(2119898+2120575)1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le 1198622(2119898+2120575)1198950

(35)

(Let us mention that intinfinminusinfin1199092119898(1 + 1199092)

120575|F(120601)(119909)|2119889119909 is finite

thanks to119873 gt 5(119898 + 120575 + 1))Combining (33) (34) and (35) we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575)1198950

1

119899 (36)

For the integer 1198950satisfying (17) it holds that

sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575+1)1198950

1

119899

le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(37)

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 5

be defined by (24) with 1198952such that 21198952 le 119899 Then there exists

a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) 1

119899)

+ 2minus21198952119904lowast)

(26)

with 119904lowast= 119904 +min(12 minus 1119901 0)

The proof follows the idea of [13 Theorem 3] and usestechnical operations on Fourier transforms

FromTheorem 4

(i) if we chose ℎ = ℎ and 1198952= 119895

0(17) we obtain

Theorem 2(ii) if ℎ and ℎ satisfy that there exist 120592 isin [0 1] and a

constant 119862 gt 0 such that

E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119909) minus ℎ (119909)

10038161003816100381610038161003816

2

119889119909) le 119862119899minus120592 (27)

then the optimal integer 1198952is such that 21198952 =

[119899120592(2119904lowast+2119898+2120575+1)] and we obtain the following rate ofconvergence for 119891(119898)

3

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast120592(2119904lowast+2119898+2120575+1)

(28)

Naturally the estimation of ℎ has a negative impact on theperformance of 119891(119898)

3 In particular if ℎ isin 119861119904

1015840

11990110158401199031015840(119872

1015840) thenthe standard density linear wavelet estimator ℎ attains therate of convergence 119899minus120592 with 120592 = 2119904

∘(2119904

∘+ 1) and 119904

∘=

1199041015840+min(12minus11199011015840 0) (and it is optimal in theminimax sensefor 1199011015840 ge 2 see Hardle et al [9]) With this choice the rate ofconvergence for 119891(119898)

3becomes 119899minus4119904lowast119904∘(2119904∘+1)(2119904lowast+2119898+2120575+1) Let

us mention that 119891(119898)3

is not adaptive since it depends on 119904However 119891(119898)

3remains an acceptable first approach for the

estimation of 119891(119898) with unknown ℎ

Conclusion and Perspectives This study considers the estima-tion of 119891(119898) from (1) According to the knowledge of ℎ ornot we propose wavelet methods and prove that they attainfast rates of convergence under the MISE over Besov ballsAmong the perspectives of this work we retain the following

(i) The relaxation of the assumption (K2) perhaps byconsidering (K21015840) there exist four constants 119862

1gt 0

120596 isin N 120578 gt 0 and 120575 ge 0 such that

1003816100381610038161003816F (119892) (119909)1003816100381610038161003816minus1

le 1198621

10038161003816100381610038161003816100381610038161003816sin(120587119909

120578)10038161003816100381610038161003816100381610038161003816

minus120596

(1 + |119909|)120575 119909 isin R (29)

This condition was first introduced by Delaigle andMeister [30] in a context of deconvolution-estimationof function It implies (K2) and has the advantage toconsider some functions 119892 having zeros in Fouriertransform domain as numerous kinds of compactlysupported functions

(ii) The construction of an adaptive version of 119891(119898)3

through the use of a thresholding method

(iii) The extension of our results to the L119901 risk with 119901 ge 1

All these aspects need further investigations that we leave forfuture works

4 Proofs

In this section 119862 denotes any constant that does not dependon 119895 119896 or 119899 Its value may change from one term to anotherand may depend on 120601 or 120595

Proof of Proposition 1 By the independence between 1198831and

1205851 E(120585

1) = 0 sup(119891 ⋆ 119892) = supp(ℎ) = [119886

lowast 119887lowast] and F(119891 ⋆

119892)(119909) = F(119891)(119909)F(119892)(119909) we have

E(1198841

ℎ (1198831)119890minus1198941199091198831) = E(

(119891 ⋆ 119892) (1198831)

ℎ (1198831)

119890minus1198941199091198831)

+ E (1205851)E(

1

ℎ (1198831)119890minus1198941199091198831)

= E((119891 ⋆ 119892) (119883

1)

ℎ (1198831)

119890minus1198941199091198831)

= int119887lowast

119886lowast

(119891 ⋆ 119892) (119910)

ℎ (119910)119890minus119894119909119910

ℎ (119910) 119889119910

= intinfin

minusinfin

(119891 ⋆ 119892) (119910) 119890minus119894119909119910

119889119910

= F (119891 ⋆ 119892) (119909)

= F (119891) (119909)F (119892) (119909)

(30)

It follows from (K1) and 119898 integration by parts that(119894119909)

119898F(119891)(119909) = F(119891(119898))(119909) Using the Fubini theorem

(119894119909)119898F(119891)(119909) = F(119891(119898))(119909) (30) and the Parseval identity

we obtain

E (119888(119898)

119895119896)

= E(1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)E(

1198841

ℎ (1198831)119890minus1198941199091198831)119889119909

6 Advances in Statistics

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)F (119891) (119909)F (119892) (119909) 119889119909

=1

2120587intinfin

minusinfin

(119894119909)119898F (119891) (119909)F (120601

119895119896) (119909) 119889119909

=1

2120587intinfin

minusinfin

F (119891(119898)) (119909)F (120601

119895119896) (119909) 119889119909

= int119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

= 119888(119898)

119895119896

(31)

Proposition 1 is proved

Proof of Theorem 2 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

0 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(32)

Using Proposition 1 (1198841 119883

1) (119884

119899 119883

119899) that are iid the

inequalitiesV(119863) le E(|119863|2) for any randomcomplex variable119863 and (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 and (K1) and (K3)we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

)

= V (119888(119898)

1198950 119896)

=1

119899V(

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

le1

(2120587)2119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le2

(2120587)2119899E(

((119891 ⋆ 119892) (1198831))2

+ 12058521

(ℎ (1198831))2

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le1

119899

2

(2120587)21198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

(33)

The Parseval identity yields

E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= int119887lowast

119886lowast

1

ℎ (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

ℎ (119910) 119889119910

le intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

F(119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

= 2120587intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

(34)

Using (K2) |F(1206011198950 119896)(119909)| = 2minus11989502|F(120601)(11990921198950)| and a change

of variables we obtain

intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

le1

11988821

intinfin

minusinfin

1199092119898(1 + 119909

2)120575 10038161003816100381610038161003816F (120601

1198950 119896) (119909)

10038161003816100381610038161003816

2

119889119909

=1

11988821

2minus1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816100381610038161003816F (120601) (

119909

21198950)1003816100381610038161003816100381610038161003816

2

119889119909

=1

11988821

intinfin

minusinfin

221198950119898119909

2119898(1 + 2

211989501199092)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le1

11988821

2(2119898+2120575)1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le 1198622(2119898+2120575)1198950

(35)

(Let us mention that intinfinminusinfin1199092119898(1 + 1199092)

120575|F(120601)(119909)|2119889119909 is finite

thanks to119873 gt 5(119898 + 120575 + 1))Combining (33) (34) and (35) we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575)1198950

1

119899 (36)

For the integer 1198950satisfying (17) it holds that

sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575+1)1198950

1

119899

le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(37)

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Statistics

=1

2120587intinfin

minusinfin

(119894119909)119898F (120601

119895119896) (119909)

F (119892) (119909)F (119891) (119909)F (119892) (119909) 119889119909

=1

2120587intinfin

minusinfin

(119894119909)119898F (119891) (119909)F (120601

119895119896) (119909) 119889119909

=1

2120587intinfin

minusinfin

F (119891(119898)) (119909)F (120601

119895119896) (119909) 119889119909

= int119887

119886

119891(119898)(119909) 120601

119895119896(119909) 119889119909

= 119888(119898)

119895119896

(31)

Proposition 1 is proved

Proof of Theorem 2 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

0 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(32)

Using Proposition 1 (1198841 119883

1) (119884

119899 119883

119899) that are iid the

inequalitiesV(119863) le E(|119863|2) for any randomcomplex variable119863 and (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 and (K1) and (K3)we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

)

= V (119888(119898)

1198950 119896)

=1

119899V(

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

(119894119909)119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909)

le1

(2120587)2119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le2

(2120587)2119899E(

((119891 ⋆ 119892) (1198831))2

+ 12058521

(ℎ (1198831))2

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le1

119899

2

(2120587)21198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

(33)

The Parseval identity yields

E(1

ℎ (1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= int119887lowast

119886lowast

1

ℎ (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

ℎ (119910) 119889119910

le intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

F(119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)) (119910)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119910

= 2120587intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896)(119909)

F(119892)(119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

(34)

Using (K2) |F(1206011198950 119896)(119909)| = 2minus11989502|F(120601)(11990921198950)| and a change

of variables we obtain

intinfin

minusinfin

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119909119898F (120601

1198950 119896) (119909)

F (119892) (119909)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

119889119909

le1

11988821

intinfin

minusinfin

1199092119898(1 + 119909

2)120575 10038161003816100381610038161003816F (120601

1198950 119896) (119909)

10038161003816100381610038161003816

2

119889119909

=1

11988821

2minus1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816100381610038161003816F (120601) (

119909

21198950)1003816100381610038161003816100381610038161003816

2

119889119909

=1

11988821

intinfin

minusinfin

221198950119898119909

2119898(1 + 2

211989501199092)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le1

11988821

2(2119898+2120575)1198950 int

infin

minusinfin

1199092119898(1 + 119909

2)120575 1003816100381610038161003816F (120601) (119909)

10038161003816100381610038162

119889119909

le 1198622(2119898+2120575)1198950

(35)

(Let us mention that intinfinminusinfin1199092119898(1 + 1199092)

120575|F(120601)(119909)|2119889119909 is finite

thanks to119873 gt 5(119898 + 120575 + 1))Combining (33) (34) and (35) we have

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575)1198950

1

119899 (36)

For the integer 1198950satisfying (17) it holds that

sum119896isinΛ 1198950

E (10038161003816100381610038161003816119888(119898)

1198950 119896minus 119888

(119898)

1198950 119896

10038161003816100381610038161003816

2

) le 1198622(2119898+2120575+1)1198950

1

119899

le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(37)

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 7

Let us now bound the last term in (32) Since 119891(119898) isin

119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we obtain

infin

sum119895=1198950

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198950119904lowast le 119862119899

minus2119904lowast(2119904lowast+2119898+2120575+1) (38)

Owing to (32) (37) and (38) we have

E(int119887

119886

10038161003816100381610038161003816119891(119898)

1(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862119899minus2119904lowast(2119904lowast+2119898+2120575+1)

(39)

Theorem 2 is proved

Proof of Theorem 3 For 120574 isin 120601 120595 any integer 119895 ge 120591 and119896 isin Λ

119895

(a1) using arguments similar to those in Proposition 1 weobtain

E(1

119899

119899

sumV=1

119884V

ℎ (119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909)

= int119887

119886

119891(119898)(119909) 120574

119895119896(119909) 119889119909

(40)

(a2) using (33) (34) and (35) with 120574 instead of 120601 we have

119899

sumV=1

E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

119884V

ℎ(119883V)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896)(119909)

F(119892)(119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

= 119899E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1198841

ℎ (1198831)

1

2120587intinfin

minusinfin

119909119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

)

le 1198622

lowast1198992

(2119898+2120575)119895

(41)

with 1198622lowast

= (1(120587119888211988821))(1198622

1(int

infin

minusinfin|119892(119909)|119889119909)

2+

E(12058521)) int

infin

minusinfin119909119898(1 + 1199092)

120575|F(120574)(119909)|2119889119909

Thanks to (a1) and (a2) we can apply [25 Theorem 61] (seeAppendix) with 120583

119899= 120592

119899= 119899 120590 = 119898 + 120575 120579

120574= 119862

lowast 119882V =

(119884V 119883V)

119902V (120574 (119910 119911)) =119910

ℎ (119911)

1

2120587intinfin

minusinfin

(119894119909)119898F (120574

119895119896) (119909)

F (119892) (119909)119890minus119894119909119911

119889119909

(42)

and 119891(119898) isin 119861119904119901119903(119872) with119872 gt 0 119903 ge 1 either 119901 ge 2 and

119904 isin (0119873) or 119901 isin [1 2) and 119904 isin (1119901119873) we prove theexistence of a constant 119862 gt 0 such that

E(int119887

119886

10038161003816100381610038161003816119891(119898)

2(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909) le 119862(ln 119899119899)

2119904(2119904+2119898+2120575+1)

(43)

Theorem 3 is proved

Proof of Theorem 4 We expand the function 119891(119898) on B as(8) at the level ℓ = 119895

2 SinceB forms an orthonormal basis of

L2([119886 119887]) we get

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

= sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952119896

10038161003816100381610038161003816

2

) +

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

(44)

Using 119891(119898) isin 119861119904119901119903(119872) sube 119861

119904lowast

2infin(119872) (see [9 Corollary 92]) we

have

infin

sum119895=1198952

sum119896isinΛ 119895

(119889(119898)

119895119896)2

le 1198622minus21198952119904lowast (45)

Let 119888(119898)1198952 119896

be (15) with 119899 = 119886119899and 119895 = 119895

2 The elementary

inequality (119909 + 119910)2 le 2(1199092 + 1199102) (119909 119910) isin R2 yields

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

) le 2 (1198781+ 119878

2) (46)

where

1198781= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

1198782= sum

119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

(47)

Upper Bound for 1198782 Proceeding as in (37) we get

1198782le 1198622

(2119898+2120575+1)11989521

119886119899

le 1198622(2119898+2120575+1)1198952

1

119899 (48)

Upper Bound for 1198781The triangular inequality gives

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

le1

(2120587) 119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

(49)

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Statistics

Owing to the triangular inequality the indicator function(K3) |ℎ(119883V)| lt 11988822 sube |ℎ(119883V) minus ℎ(119883V)| gt 11988822 and theMarkov inequality we have

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)1|ℎ(119883V)|ge11988822

minus1

ℎ (119883V)

1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

1

ℎ (119883V)((

ℎ (119883V) minus ℎ (119883V)

ℎ (119883V)) 1

|ℎ(119883V)|ge11988822

minus 1|ℎ(119883V)|lt11988822

)

1003816100381610038161003816100381610038161003816100381610038161003816

le1

ℎ (119883V)(2

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816+ 1

|ℎ(119883V)minusℎ(119883V)|gt11988822)

le4

1198882

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(50)

Therefore

10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816le 119862119860

1198952 119896119899 (51)

where

119860119895119896119899

=1

119886119899

119886119899

sumV=1

1003816100381610038161003816119884V1003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119883V119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883V) minus ℎ (119883V)

10038161003816100381610038161003816

ℎ (119883V)

(52)

Let us now consider 119880119899= (119883

119886119899+1 119883

119899) For any complex

random variable119863 we have the equality

E (1198632) = E (E (119863

2| 119880

119899))

= E (V (119863 | 119880119899)) + E ((E (119863 | 119880

119899))2

)

(53)

where E(119863 | 119880119899) denotes the expectation of 119863 conditionally

to 119880119899and V(119863 | 119880

119899) and the variance of 119863 conditionally to

119880119899 Therefore

1198781le 119862 sum

119896isinΛ 1198952

E (1198602

1198952 119896119899) = 119862 (119882

1198952 119899+ 119885

1198952 119899) (54)

where

1198821198952 119899

= sum119896isinΛ 1198952

E (V (1198601198952 119896119899

| 119880119899))

1198851198952 119899

= sum119896isinΛ 1198952

E ((E (1198601198952 119896119899

| 119880119899))

2

)

(55)

Let us now observe that owing to the independence of (1198841

1198831) (119884

119899119883

119899) the randomvariables |119884

1||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)

F(119892)(119909))119890minus1198941199091198831 119889119909||ℎ(1198831) minus ℎ(119883

1)|ℎ(119883

1)

|119884119886119899||int

infin

minusinfin119909119898(F(120601

1198952 119896)(119909)F(119892)(119909))119890minus119894119909119883119886119899119889119909||ℎ(119883

119886119899) minus ℎ(119883

119886119899)|

ℎ(119883119886119899) conditionally to 119880

119899are independent Using this

property with the inequalities V(119863 | 119880119899) le E(1198632 | 119880

119899) for

any complex random variable 119863 and (119909 + 119910)2 le 2(1199092 + 1199102)(119909 119910) isin R2 the independence between 119883

1and 120585

1 (K1) and

(K3) we get

V (1198601198952 119896119899

| 119880119899)

=1

119886119899

V(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le1

119886119899

E(1198842

1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

100381610038161003816100381610038161003816100381610038161003816

ℎ(1198831) minus ℎ(119883

1)

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899)

le1

119886119899

2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot E(

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ (1198831)

| 119880119899)

=2

1198882

(1198622

1(int

infin

minusinfin

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909)

2

+ E (1205852

1))

sdot1

119886119899

int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ (119910)ℎ (119910) 119889119910

le 1198621

119899int119887lowast

119886lowast

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

sdot10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910

(56)

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 9

Owing to (K2) |F(1206011198952 119896)(119909)| = 2minus11989522|F(120601)(11990921198952)| and a

change of variables we obtain

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus119894119909119910

119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

le intinfin

minusinfin

|119909|119898

10038161003816100381610038161003816F (120601

1198952 119896) (119909)

100381610038161003816100381610038161003816100381610038161003816F (119892) (119909)

1003816100381610038161003816119889119909

le1

1198881

intinfin

minusinfin

|119909|119898(1 + 119909

2)1205752 10038161003816100381610038161003816

F (1206011198952 119896) (119909)

10038161003816100381610038161003816119889119909

=1

1198881

2minus11989522 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816100381610038161003816

F (120601) (119909

21198952)1003816100381610038161003816100381610038161003816119889119909

=1

1198881

211989522 int

infin

minusinfin

21198952119898 |119909|

119898(1 + 2

211989521199092)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le1

1198881

2(119898+120575+12)1198952 int

infin

minusinfin

|119909|119898(1 + 119909

2)1205752 1003816100381610038161003816F (120601) (119909)

1003816100381610038161003816 119889119909

le 1198622(119898+120575+12)1198952

(57)

Therefore using Card(Λ1198952) le 11986221198952 and 21198952 le 119899 we obtain

1198821198952 119899

le 1198622(2119898+2120575+1)11989522

11989521

119899E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

(58)

Now by the Holder inequality for conditional expectationsarguments similar to (33) (34) and (35) we get

E (1198601198952 119896119899

| 119880119899)

= E(100381610038161003816100381611988411003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

sdot

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

ℎ (1198831)

| 119880119899)

le (E(11988421

ℎ (1198831)

sdot

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896) (119909)

F (119892) (119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

| 119880119899))

12

sdot (E(

10038161003816100381610038161003816ℎ (119883

1) minus ℎ (119883

1)10038161003816100381610038161003816

2

ℎ(1198831)

| 119880119899))

12

= (E(11988421

ℎ(1198831)

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

intinfin

minusinfin

119909119898F (120601

1198952 119896)(119909)

F(119892)(119909)119890minus1198941199091198831119889119909

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2

))

12

sdot (int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

ℎ(119910)ℎ(119910)119889119910)

12

le 1198622(119898+120575)1198952 (int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910)

12

(59)

Hence

1198851198952 119899

le 1198622(2119898+2120575+1)1198952E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (60)

It follows from (54) (58) and (60) that

1198781le 1198622

(2119898+2120575+1)1198952E(int119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) (61)

Putting (46) (48) and (61) together we get

sum119896isinΛ 1198952

E (10038161003816100381610038161003816119888(119898)

1198952 119896minus 119888

(119898)

1198952 119896

10038161003816100381610038161003816

2

)

le 1198622(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

(62)

Combining (44) (45) and (62) we obtain the desiredresult that is

E(int119887

119886

10038161003816100381610038161003816119891(119898)

3(119909) minus 119891

(119898)(119909)10038161003816100381610038161003816

2

119889119909)

le 119862(2(2119898+2120575+1)1198952 max(E(int

119887lowast

119886lowast

10038161003816100381610038161003816ℎ (119910) minus ℎ (119910)

10038161003816100381610038161003816

2

119889119910) 1

119899)

+2minus21198952119904lowast)

(63)

Theorem 4 is proved

Appendix

Let us now present in detail the general result of [25Theorem61] used in the proof of Theorem 3

We consider the wavelet basis presented in Section 2 anda general form of the hard thresholding wavelet estimatordenoted by 119891

119867for estimating an unknown function 119891 isin

L2([119886 119887]) from 119899 independent random variables1198821 119882

119899

119891119867(119909) = sum

119896isinΛ 120591

120591119896120601120591119896(119909) +

1198951

sum119895=120591

sum119896isinΛ 119895

1205731198951198961|120573119895119896|ge120581120599119895

120595119895119896(119909)

(A1)

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Advances in Statistics

where

119895119896=1

120592119899

119899

sum119894=1

119902119894(120601

119895119896119882

119894)

120573119895119896=1

120592119899

119899

sum119894=1

119902119894(120595

119895119896119882

119894) 1

|119902119894(120595119895119896 119882119894)|le120589119895

120589119895= 120579

1205952120590119895 120592

119899

radic120583119899ln 120583

119899

120599119895= 120579

1205952120590119895radic

ln 120583119899

120583119899

(A2)

120581 ge 2 + 83 + 2radic4 + 169 and 1198951is the integer satisfying

21198951 = [120583

12120590+1

119899] (A3)

Here we suppose that there exist

(i) 119899 functions 1199021 119902

119899with 119902

119894 L2([119886 119887]) times 119882

119894(Ω) rarr

C for any 119894 isin 1 119899

(ii) two sequences of real numbers (120592119899)119899isinN and (120583

119899)119899isinN

satisfying lim119899rarrinfin

120592119899= infin and lim

119899rarrinfin120583119899= infin

such that for 120574 isin 120601 120595

(1198601) any integer 119895 ge 120591 and any 119896 isin Λ119895

E(1

120592119899

119899

sum119894=1

119902119894(120574

119895119896119882

119894)) = int

119887

119886

119891 (119909) 120574119895119896(119909) 119889119909 (A4)

(1198602) there exist two constants 120579120574gt 0 and 120590 ge 0 such that

for any integer 119895 ge 120591 and any 119896 isin Λ119895

119899

sum119894=1

E (10038161003816100381610038161003816119902119894(120574119895119896119882

119894)10038161003816100381610038161003816

2

) le 1205792

120574221205901198951205922119899

120583119899

(A5)

Let 119891119867

be (A1) under (1198601) and (1198602) Suppose that 119891 isin

119861119904119901119903(119872) with 119903 ge 1 119901 ge 2 and 119904 isin (0119873) or 119901 isin [1 2)

and 119904 isin ((2120590 + 1)119901119873) Then there exists a constant 119862 gt 0such that

E(int119887

119886

10038161003816100381610038161003816119891119867(119909) minus 119891 (119909)

10038161003816100381610038161003816

2

119889119909) le 119862(ln 120583

119899

120583119899

)

2119904(2119904+2120590+1)

(A6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are thankful to the reviewers for their commentswhich have helped in improving the presented work

References

[1] L Cavalier and A Tsybakov ldquoSharp adaptation for inverseproblems with random noiserdquo Probability Theory and RelatedFields vol 123 no 3 pp 323ndash354 2002

[2] M Pensky and T Sapatinas ldquoOn convergence rates equivalencyand sampling strategies in functional deconvolution modelsrdquoThe Annals of Statistics vol 38 no 3 pp 1793ndash1844 2010

[3] J-M Loubes and C Marteau ldquoAdaptive estimation for aninverse regression model with unknown operatorrdquo Statistics ampRisk Modeling vol 29 no 3 pp 215ndash242 2012

[4] N Bissantz and M Birke ldquoAsymptotic normality and confi-dence intervals for inverse regressionmodels with convolution-type operatorsrdquo Journal of Multivariate Analysis vol 100 no 10pp 2364ndash2375 2009

[5] M Birke N Bissantz and H Holzmann ldquoConfidence bandsfor inverse regression modelsrdquo Inverse Problems vol 26 no 11Article ID 115020 2010

[6] T Hildebrandt N Bissantz and H Dette ldquoAdditive inverseregression models with convolution-type operatorsrdquo ElectronicJournal of Statistics vol 8 no 1 pp 1ndash40 2014

[7] B L S Prakasa Rao Nonparametric Functional EstimationAcademic Press Orlando Fla USA 1983

[8] A Antoniadis ldquoWavelets in statistics a review (with discus-sion)rdquo Journal of the Italian Statistical Society Series B vol 6pp 97ndash144 1997

[9] W Hardle G Kerkyacharian D Picard and A TsybakovWavelets Approximation and Statistical Applications vol 129 ofLectures Notes in Statistics Springer New York NY USA 1998

[10] B Vidakovic Statistical Modeling by Wavelets John Wiley ampSons New York NY USA 1999

[11] T T Cai ldquoOn adaptive wavelet estimation of a derivative andother related linear inverse problemsrdquo Journal of StatisticalPlanning and Inference vol 108 no 1-2 pp 329ndash349 2002

[12] A Petsa and T Sapatinas ldquoOn the estimation of the functionand its derivatives in nonparametric regression a Bayesiantestimation approachrdquo Sankhya A vol 73 no 2 pp 231ndash2442011

[13] C Chesneau ldquoA note on wavelet estimation of the derivatives ofa regression function in a random design settingrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2014Article ID 195765 8 pages 2014

[14] B Delyon and A Juditsky ldquoOn minimax wavelet estimatorsrdquoApplied Computational Harmonic Analysis vol 3 no 3 pp 215ndash228 1996

[15] C Chesneau ldquoWavelet estimation of the derivatives of anunknown function from a convolution modelrdquo Current Devel-opment in Theory and Applications of Wavelets vol 4 no 2 pp131ndash151 2010

[16] M Pensky and B Vidakovic ldquoOn non-equally spaced waveletregressionrdquoAnnals of the Institute of StatisticalMathematics vol53 no 4 pp 681ndash690 2001

[17] A Cohen I Daubechies and P Vial ldquoWavelets on the intervaland fast wavelet transformsrdquo Applied and Computational Har-monic Analysis vol 1 no 1 pp 54ndash81 1993

[18] I Daubechies Ten Lectures on Wavelets SIAM 1992[19] S Mallat A Wavelet Tour of Signal Processing ElsevierAca-

demic Press Amsterdam The Netherlands 3rd edition 2009[20] Y MeyerWavelets and Operators Cambridge University Press

Cambridge UK 1992

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Statistics 11

[21] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoDensity estimation by wavelet thresholdingrdquo The Annals ofStatistics vol 24 no 2 pp 508ndash539 1996

[22] A B Tsybakov Introduction a lrsquoEstimation Non ParametriqueSpringer Berlin Germany 2004

[23] J Fan and J-YKoo ldquoWavelet deconvolutionrdquo IEEETransactionson Information Theory vol 48 no 3 pp 734ndash747 2002

[24] M Pensky and B Vidakovic ldquoAdaptive wavelet estimator fornonparametric density deconvolutionrdquoThe Annals of Statisticsvol 27 no 6 pp 2033ndash2053 1999

[25] Y P Chaubey C Chesneau and H Doosti ldquoAdaptive waveletestimation of a density from mixtures under multiplicativecensoringrdquo Statistics A Journal of Theoretical and AppliedStatistics 2014

[26] D L Donoho and I M Johnstone ldquoIdeal spatial adaptation bywavelet shrinkagerdquo Biometrika vol 81 no 3 pp 425ndash455 1994

[27] D L Donoho and I M Johnstone ldquoAdapting to unknownsmoothness via wavelet shrinkagerdquo Journal of the AmericanStatistical Association vol 90 no 432 pp 1200ndash1224 1995

[28] D L Donoho IM Johnstone G Kerkyacharian andD PicardldquoWavelet shrinkage asymptopiardquo Journal of the Royal StatisticalSociety Series B Methodological vol 57 no 2 pp 301ndash369 1995

[29] A Juditsky and S Lambert-Lacroix ldquoOn minimax densityestimation on Rrdquo Bernoulli Official Journal of the BernoulliSociety for Mathematical Statistics and Probability vol 10 no2 pp 187ndash220 2004

[30] A Delaigle and A Meister ldquoNonparametric function estima-tion under Fourier-oscillating noiserdquo Statistica Sinica vol 21no 3 pp 1065ndash1092 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of