robust extrapolation problem for stochastic sequences · pdf filethe mean square optimal...

Columbia International Publishing Contemporary Mathematics and Statistics (2013) Vol. 1 No. 3 pp. 123-150 doi:10.7726/cms.2013.1009

Research Article

______________________________________________________________________________________________________________________________ *Corresponding e-mail: [email protected] 1* Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National

University of Kyiv, Kyiv 01601, Ukraine 2 Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National

University of Kyiv, Kyiv 01601, Ukraine 123

Robust Extrapolation Problem for Stochastic Sequences with Stationary Increments

Mikhail Moklyachuk 1* and Maksym Luz 2

Received 17 June 2013; Published online 16 November 2013 © The author(s) 2013. Published with open access at www.uscip.us

Abstract

The problem of optimal estimation of functionals )()(=0=

kkaAk

and )()(=0=

kkaAN

kN which

depend on the unknown values of stochastic sequence )(k with stationary n th increments is considered.

Estimates are based on observations of the sequence )(m at points of time 2,1,= m . Formulas for

calculating the value of the mean square error and the spectral characteristic of the optimal linear estimates of the functionals are derived in the case where spectral density of the sequence is exactly known. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the sequence is not known but a set of admissible spectral densities is given. Keywords: Stochastic sequence with stationary increments; Minimax-robust estimate; Mean square error; Least favorable spectral density; Minimax-robust spectral characteristic

1. Introduction

Stochastic processes with n th stationary increments ),()( tn , ,t R , were introduced by

Yaglom (1955). He described the main properties of these processes, found the spectral representation of stationary increments and solved the extrapolation problem for processes with stationary increments. Further results for such stochastic processes were presented by Pinsker (1955), Yaglom and Pinsker (1954). See Yaglom (1987a, 1987b) for more relative results and references.

Mikhail Moklyachuk and Maksym Luz / Contemporary Mathematics and Statistics (2013) Vol. 1 No. 3 pp. 123-150

124

The mean square optimal estimation problems for stochastic processes with n th stationary increments are natural generalization of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes. Traditional methods of solution of the linear extrapolation, interpolation and filtering problems for stationary stochastic processes were developed by A.N. Kolmogorov, N.Wiener, A.M.Yaglom (see, for example, selected works of Kolmogorov (1992), survey article by Kailath (1974), books by Rozanov (1967), Wiener (1966), Yaglom (1987a, 1987b)). These methods are based on the assumption that the spectral density of the process is known. In practice, however, it is impossible to have complete information on the spectral density in most cases. To solve the problem one finds parametric or nonparametric estimates of the unknown spectral density or selects a density by other reasoning. Then the classical estimation method is applied provided that the estimated or selected density is the true one. This procedure can result in significant increasing of the value of error as Vastola and Poor (1983) have demonstrated with the help of some examples. This is a reason to search estimates which are optimal for all densities from a certain class of admissible spectral densities. These estimates are called minimax since they minimize the maximal value of the error. A survey of results in minimax (robust) methods of data processing can be found in the paper by Kassam and Poor (1985). The paper by Ulf Grenander (1957) should be marked as the first one where the minimax extrapolation problem for stationary processes was formulated and solved. Franke and Poor (1984), Franke (1985) investigated the minimax extrapolation and filtering problems for stationary sequences with the help of convex optimization methods. This approach makes it possible to find equations that determine the least favorable spectral densities for various classes of admissible densities. For more details see, for example, books by Kurkin et al. (1990), Moklyachuk (2008), Moklyachuk and Masyutka (2012). In papers by Moklyachuk (1994-2008) the minimax approach was applied to extrapolation, interpolation and filtering problems for functionals which depend on the unknown values of stationary processes and sequences. Methods of solution the minimax-robust estimation problems for vector-valued stationary sequences and processes were developed by Moklyachuk and Masyutka (2006-2011). The minimax-robust estimation problems (extrapolation, interpolation and filtering) for linear functionals which depend on unknown values of periodically correlated stochastic processes were investigated by Dubovets'ka and Moklyachuk (2012-2013). Luz and Moklyachuk (2012a, 2012b) investigated the minimax interpolation problem for the linear

functional )()(=0=

kkaAN

kN which depends unknown values of a stochastic sequence )(m

with stationary n th increments from observations of the sequence at points \{0,1, , }NKZ .

In this article we focus on the mean square optimal estimates of the functionals

)()(=),()(=0=0=

kkaAkkAN

k

N

k

(1)

which depend on the unknown values of a stochastic sequence )(k with stationary n th

increments. Estimates are based on observations of the sequence )(m at points 2,1,= m .

The estimation problem for sequences with stationary increments is solved in the case of spectral certainty where the spectral density of the sequence is exactly known as well as in the case of


125

spectral uncertainty where the spectral density of the sequence is not known but a set of admissible spectral densities is given. Formulas are derived for computing the value of the mean-square error

and the spectral characteristic of the optimal linear estimates of functionals A and NA in the

case of spectral certainty. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case of spectral uncertainty for concrete classes of admissible spectral densities.

2. Stationary stochastic increment sequence. Spectral representation

Definition 2.1 For a given stochastic sequence { ( ), }m m Z a sequence

),(1)(=)()(1=),(0=

)( lmCmBm l

n

ln

l

nn (2)

where B is a backward shift operator with step Z , such that )(=)( mmB , is called the

stochastic n th increment sequence with step Z .

For the stochastic n th increment sequence ),()( mn the following relations hold true:

),,(1)(=),( )()( nmm nnn (3)

,),,(=),( )(1)(

0=

)( N

klmAkm n

l

nk

l

n (4)

where coefficients }1)(,0,1,2,=,{ nklAl are determined by the representation

.=)(11)(

0=

1 l

l

nk

l

nk xAxx

Definition 2.2 The stochastic n th increment sequence ),()( mn generated by stochastic sequence

{ ( ), }m m Z is wide sense stationary if the mathematical expectations

)(=),(E )(

0

)( nn cm

and

),,(=),(),(E 21

)(

20

)(

10

)( mDmmm nnn

exist for all 210 ,,,, mm and do not depend on 0m . The function )()( nc is called the mean value

of the n th increment sequence and the function ),,( 21

)( mD n is called the structural function of

the stationary n th increment sequence (or the structural function of n th order of the stochastic sequence { ( ), }m m Z ).


126

The stochastic sequence { ( ), }m m Z which determines the stationary n th increment sequence

),()( mn by formula (2) is called sequence with stationary n th increments.

Theorem 2.1 The mean value )()( nc and the structural function ),,( 21

)( mD n of the stochastic

stationary n th increment sequence ),()( mn can be represented in the following forms

,=)()( nn cc (5)

),(1

)(1)(1=),,(2

2121

)(

dFeeemDn

ninimin

(6)

where c is a constant, )(F is a left-continuous nondecreasing bounded function with .0=)( F

The constant c and the function )(F are determined uniquely by the increment sequence

),()( mn .

From the other hand, a function )()( nc which has the form (5) with a constant c and a function

),,( 21

)( mD n which has the form (6) with a function )(F which satisfies the indicated conditions

are the mean value and the structural function of some stationary n th increment sequence

),()( mn .

Using representation (6) of the structural function of a stationary n th increment sequence

),()( mn and the Karhunen theorem (see Karhunen (1947)), we get the following spectral

representation of the stationary n th increment sequence ),()( mn :

),()(

1)(1=),()(

dZi

eemn

nimin

(7)

where )(Z is an orthogonal stochastic measure оn ),[ connected with the spectral function

)(F by the relation

).(=)()(E 2121 AAFAZAZ (8)

Denote by )( )(nH the Hilbert space generated by all elements ( ){ ( , ) : , }n m m Z in the space

2( , , )H L F P and let )( )(ntH , tZ , be the subspace of )( )(nH generated by elements

0}>,:),({ )( tmmn . Let

).(=)( )()( nt

t

n HS Z


127

Since the space )( )(nS is a subspace in the Hilbert space )( )(nH , the space )( )(nH admits the

decomposition

),()(=)( )()()( nnn RSH

where )( )(nR is an orthogonal complement of the subspace )( )(nS in the space )( )(nH .

Definition 2.3 A stationary n th increment sequence ),()( mn is called regular if

)(=)( )()( nn RH . It is called singular if )(=)( )()( nn SH .

Theorem 2.2 A wide-sense stationary stochastic increment sequence admits a unique representation in the form

),,(),(=),( )()()( mmm n

s

n

r

n (9)

where ( ){ ( , ) : }n

r m m Z is a regular increment sequence and ( ){ ( , ) : }n

s m m Z is a singular

increment sequence. Moreover, the increment sequences ( ) ( , )n

r m and ),()( kn

s are orthogonal

for all ,m kZ .

Components of representation (9) are constructed in the following way

).,(),(=),()],(|),([E=),( )()()()()()( mmmSmm n

s

nn

r

nnn

s

Let { : }m m Z be a sequence of uncorrelated random variables with 0=E m and 1=D 2

m .

Define the Hilbert space )(tH generated by elements }:{ tmm .

Definition 2.4 A sequence of uncorrelated random variables { : }m m Z is called innovation

sequence for a regular stationary n th increment sequence ),()( mn if the condition

)(=)( )( tnt HH holds true for all tZ .

Theorem 2.3 A stochastic stationary increment sequence ),()( mn is regular if and only if there

exists an innovation sequence { : }m m Z and a sequence of complex functions 0}:),({ )( mkn

,

<|),(| 2)(

0= kn

k, such that

).(),(=),( )(

0=

)( kmkm n

k

n

(10)

Representation (10) is called canonical moving average representation of the stochastic stationary

increment sequence ),()( mn .


128

Corollary 2.1 A wide-sense stationary stochastic increment sequence admits a unique representation in the form

),(),(),(=),( )(

0=

)()( kmkmm n

k

n

s

n

(11)

where

<|),(| 2)(

0= kn

k and { : }m m Z is the innovation sequence.

Let the stationary n th increment sequence ),()( mn admit the canonical representation (10). In

this case the spectral function )(F of the stationary increment sequence ),()( mn has a spectral

density )(f which admits the canonical factorization

,)(=)(,|)(=|)(0=

2 k

k

i zkzef

(12)

where the function k

kzkz )(=)(

0=

has the convergence radius 1>r and does not have zeros

in the unit disk 1}|:|{ zz . Let us define

,)(=),(=)(0=

)(

0=

k

k

kn

k

zkzkz

where ),(=)( )( kk n are coefficients which determine the canonical representation (10) .

Then the following relation holds true

).(|1|

=)(2

22

fe

en

nii

(13)

The one-sided moving average representation (10) and relation (13) are used for finding the optimal mean square estimate of the unknown values of a sequence with n th stationary increment.

3. Hilbert space projection method of extrapolation of linear functionals

Let { ( ), }m m Z be a stochastic sequence which determines a stationary n th increment sequence

),()( mn with an absolutely continuous spectral function )(F which has spectral density )(f .

Without loss of generality we will assume that the mean value of the increment sequence

),()( mn is 0. Let the stationary increment sequence ),()( mn admit the one-sided moving

average representation (10) and the spectral density )(f admits the canonical factorization

(12) . Consider the case where the step 0> .


129

Suppose that observations of the sequence )(m at points 1, 2,...m are known. The problem

is to find the mean square optimal linear estimates of functionals )()(=0=

kkaAN

kN and

)()(=0=

kkaAk

which depend on unknown values )(m , 0m of the sequence )(m .

From (2) we can obtain the formal equation

),,()(=),()(1

1=)( )(

=

)(

jjkdkB

k nk

j

n

n

(14)

where coefficients

0}:)({ kkd are determined by the relation

.=)(0=0=

n

j

j

k

k

xxkd

From (2) and (14) one can obtain the following relations

),,()()()()(=)()( )(

=0=

1

=0=

iikdkaiivkka n

ikinik

),(1)()()(1)()(=),()(0=0=

=

1

=

)(

0= '

ilbCiilbCikkb l

n

ln

li

l

n

ln

il

ni

n

k

where ][ x denotes the least integer number among numbers which are greater or equal to x .

Using these relations we obtain representation of the functional A as difference VBA =

of functionals, where

),()(=),,()(=1

=

)(

0=

kkvVkkbBnk

n

k

,,2,1,=),(1)(=)('

=

nkklbCkv l

n

ln

kl

(15)

0.,)(=)()(=)(=

kaDkmdmakb k

km

(16)

Here D is a linear operator in the space 2 determined by elements )(=, kjdD jk if

jk 0 , and 0=,

jkD if kj < ; the vector = ( (0), (1), (2), )Ta a a a K .

We will suppose that the following restirictions on the coefficients 0}:)({ kkb hold true


130

.<|)(|1)(,|<)(| 2

0=0=

kbkkbkk

(17)

Under these conditions the functional B has the second moment and the operator B defined

below is compact. Since coefficients )(ka and )(kb are related by (16) , the following conditions

hold true

.<|)(|1)(,|<)(| 2

0=0=

k

k

k

k

aDkaD (18)

Let A denote the mean square optimal linear estimate of the functional A from observations of

the sequence )(m at points 1, 2,...m and let B denote the mean square optimal linear

estimate of the functional B from observations of the stochastic n th increment sequence

),()( mn at points 1, 2,...m . Let 2|ˆ|E:=)ˆ,( AAAf denote the mean square error

of the estimate A and let 2|ˆ|E:=)ˆ,( BBBf denote the mean square error of the

estimate B . Since values of the sequence )(m are known for nm ,2,1,= , the following

equality holds true

.ˆ=ˆ VBA (19)

From this relation we get

).ˆ,(=|ˆ|E=|ˆ|E=|ˆ|E=)ˆ,( 222 BfBBBVAAAAf

Denote by )(0

2 fL the subspace of the Hilbert space )(2 fL generated by the set of functions

1}:)(

1)(1{ k

iee

n

niki

. Every linear estimate B of the functional B can be represented

in the form

),()(

1))(1(=ˆ

dZ

iehB

n

ni

(20)

where )(h is the spectral characteristic of the estimate B . The spectral characteristic of the

optimal estimate provides the minimum value of the mean square error )ˆ,( Bf .

With the help of the Hilbert space projection method proposed by Kolmogorov we can find formulas for calculation the mean square error and the spectral characteristic of the optimal linear

estimate B of the functional B . Following the method we find that the the spectral

characteristic )(h of the optimal linear estimate is determined by the following conditions:


131

1) fLi

ehn

ni 0

2)(

1))(1(

;

2) fLi

eheBn

nii 0

2)(

1)))(1()((

, where

.)(=)(0=

ki

k

i ekbeB

From the second condition we obtain the following relation for every 1k

0.=)(1

|1|))()((2

2

dfeeheB ki

n

nii

These relations are satisfied by the function

),()()(=)( 1

iii eereBh (21)

,)(=)()(=)(0=0=0=

ji

j

j

ji

mj

i eBemjmber

where B is a linear symmetric operator in the space 2 defined by the matrix with elements

)(=, jkbB jk , 0, jk . )(2),(1),(0),(= ; )(k , 0k , are coefficients which

determine the moving average representation (10).

Note that under conditions (17) the operator B is compact.

To check condition 1) it is sufficient to show that the function 0

2)( Lh , where 0

2L is the closed

linear subspace of the space ),(2 L generated by the set of functions 1}:{ ke ki . Since 0

2

1 )( Le i

, we have

=)())()()((=)( 1

iiii eereeBh

.)()()(= 0

2

=

1

=

1

Lemjmbe ji

jmj

i

Therefore the spectral characteristic )(=:)( fhh of the optimal estimate B of the functional

B can be calculated by formula (21).

The value of the mean square error )ˆ,( Bf can be calculated by the formula


132

.||=|||)(|2

1=)ˆ,( 22

BderBf i

(22)

Summarizing our reasoning we have the following theorem. Theorem 3.1 Let a stochastic sequence { ( ), }m m Z determine a stationary stochastic n th

increment sequence ),()( mn with absolutely continuous spectral function )(F and spectral

density )(f which admits the canonical factorization (12) . The optimal linear estimate B of the

functional B which depends on the unobserved values ),()( mn , 0,1,2,=m , 0> , from

observations of the sequence )(m at points 2,1,= m , can be calculated by formula (20). The

spectral characteristic )(h of the optimal linear estimate B can be calculated by formula (21) .

The value of the mean square error )ˆ,( Bf can be calculated by formula (22) .

Using Theorem 3.1 and representation (9) , we can obtain the optimal estimate of an unobserved

value of the sequence ),()( mn , 0m , from observations of the sequence )(k at points

2,1,= k The singular component ),()( kn

s of the sequence has errorless estimate. We will

use formula (21) to obtain the spectral characteristic )(, mh of the optimal estimate ),(ˆ )( mn

of the regular component ),()( kn

r of the sequence. Consider the vector b with 1 on position m ,

0m , and 0 on other positions. It follows from the derived formulas that the spectral characteristic of the estimate

)(

)(

1))(1(),(=),(ˆ

,

)()( dZi

ehkmn

ni

m

n

s

n (23)

can be calculated by the formula

.)()(=)(0=

1

,

kim

k

imi

m ekeeh

(24)

The value of the mean square error can be calculated by the formula

.|)(|=)(2

1=)),(ˆ,( 2

0=

2

0=

)( kdekmfm

k

kim

k

n

(25)

The following statement holds true.


133

Corollary 3.1 The optimal linear estimate ),(ˆ )( mn of the value of the increment sequence

),()( mn , 0m , 0> , from observations of the sequence )(k at points 2,1,= k can be

calculated by formula (23) . The spectral characteristic )(, mh of the optimal linear estimate

),(ˆ )( mn can be calculated by formula (24) . The value of mean square error )),(ˆ,( )( mf n of

the optimal linear estimate can be calculated by formula (25) .

Making use relation (19) we can find the optimal estimate A of the functional A from

observations of the sequence )(k at points 2,1,= k . These estimate can be presented in the

following form

,)()(

1))(1()()(=ˆ )(

1

=

dZi

ehkkvAn

nia

nk

(26)

where coefficients )(kv for nk ,2,1,= are defined by relation (15) . Using relationship

(16) between coefficients )(ka and )(kb , we obtain the following equation

,)(=)()()(=)(=0=

k

klm

k ADkldlmamB

where the linear operator A is defined by coefficients )(ka , 0k , in the following way:

)(=)( , jkaA jk , 0, jk . Thus the spectral characteristic and the value of the mean square error

of the optimal estimate A can be calculated by the formulas

),()()(=)( 1)()(

iiaia eereAh (27)

.)(=)(,)(=)(0=

)(

0=

ji

j

j

iaki

k

k

i eADereaDeA

(28)

.||=|||)(|2

1=)ˆ,( 22)(

ADderAf ia

(29)

The following theorem holds true. Theorem 3.2 Let a stochastic sequence { ( ), }m m Z determine a stationary stochastic n th


density )(f which admits the canonical factorization (12) . The optimal linear estimate A of the

functional A of unobserved values )(m , 0,1,2,=m , from observations of the sequence )(m at

points 2,1,= m , can be calculated by formula (26) . The spectral characteristic )()( ah of the


134

optimal linear estimate A can be calculated by formula (27) . The value of the mean square error

)ˆ,( Af of the optimal linear estimate can be calculated by formula (29) .

Consider now the problem of the mean square optimal estimation of the functional NA .

Using the derived formulas we can find the optimal estimate of the functional NA in the form

,)()(

1))(1()()(=ˆ )(

,,

1

=

dZi

ehkkvAn

nia

NN

nk

N (30)

where coefficients )(, kv N , nk ,2,1,= , are calculated by formulas

,,2,1,=),(1)(=)( ,

,min

=

, nkklbCkv N

l

n

l

nkN

kl

N

.,0,1,=,)(=)()(=)(=

, NkaDkmdmakb kNN

N

km

N

Here ND is the matrix of dimension 1)(1)( NN with elements )(=, kjdD jk

if

Njk 0 , and 0=,

jkD if kj < or Nkj >, ; ))(,(2),(1),(0),(= NaaaaaN . The spectral

characteristic of the optimal estimate NA can be calculated by the following formulas:

),()()(=)( 1)(

,

)(

,

iia

N

i

N

a

N eereAh (31)

,)(=)(,)(=)( ,

0=

)(

,

0=

ji

jNNN

N

j

ia

N

ki

kNN

N

k

i

N eADereaDeA

(32)

where the matrix NA of dimension 1)(1)( NN is determined by coefficients )(ka ,

Nk ,0,1,= , in the following way: )(=)( , jkaA jkN if Njk 0 , 0=)( , jkNA if Njk > ,

Njk ,0 . The value of the mean square error of the optimal estimate NA can be calculated by

the following formula:

.||=|||)(|2

1=|ˆ|E:=)ˆ,( 2

,

2)(

,

2

NNN

ia

NNNN ADderAAAf

(33)

Consequently, the following theorem holds true. Theorem 3.3 Let a stochastic sequence { ( ), }m m Z determine a stationary stochastic n th



135

density )(f which admits the canonical factorization (12) . The optimal linear estimate NA of the

functional NA of unobserved values )(m , 0,1,2,=m , from observations of the sequence )(m

at points 2,1,= m can be calculated by formula (30) . The spectral characteristic )()(

, a

Nh of the

optimal linear estimate NA can be calculated by formula (31) . The value of mean square error

)ˆ,( NAf can be calculated by formula (33) .

Consider the case where 0> m . In this case the mean square optimal estimate of the value

)(m , 0m , can be calculated by formula

)()(

1))(1()(1)(=)(ˆ ,

=1

dZi

ehlmCmn

ni

m

l

n

ln

l

(34)

The spectral characteristic )(, mh and the value of the mean square error

)),(ˆ,(=))(ˆ,( )( mfmf n of the estimate of the element )(m can be calculated by formulas

(24) and (25) respectively.

Consequently, the following statement holds true.

Corollary 3.2 Let 0> m . The optimal mean square estimate )(ˆ m of the element )(m ,

0> m , from observations of the sequence )(m at points 2,1,= m can be calculated by

formula (34) . The spectral characteristic )(, mh of the optimal linear estimate )(ˆ m can be

calculated by formula (24) . The value of mean square error ))(ˆ,( mf can be calculated by

formula (25) .

Remark 3.1 Using relation (13) we can find a relationship between coefficients

}0,1,2,=:)({ kk and }0,1,2,=:)({ kk . So far as

<|1|

ln2

2

de

n

ni

for every 1n and 1 , there is a function k

kzkwzw )(=)(

0=

such that

<|)(| 2

0=kw

k ,

2

2

2

|)(=||1|

i

n

ni

ewe

and the following representation holds true:

).()(=)(

iii eewe (35)

The function )(zw is determined by the relation


136

.|1|

ln4

1=)(

2

2

de

ze

zeExpzw

n

ni

i

i

(36)

From (35) we can get

).()(=)(0=

jjkwkk

j

Therefore elements )(2),(1),(0),(= and )(2),(1),(0),(= from the space 2

are connected by the following relation

,= W (37)

where W is a linear operator in the space 2 with elements )(=, kjwW kj if jk 0 and

0=,

kjW if .< kj The vectors ))(,(2),(1),(0),(=, NN and

))(,(2),(1),(0),(= NN are connected by the relation

,=, NNN W (38)

where

NW ia a matrix of dimension 1)(1)( NN with elements )(=, kjwW kj if

Njk 0 and 0=,

kjW if kj < , Nkj ,0,1,=, .

Example 3.1 Consider an (0,1,1)ARIMA sequence { ( ) : }m m Z . Increments of order 1 of the

sequence )(m are stationary and increments with step 1 form one-sided moving average

stochastic sequence of order 1 with parameter . The spectral density of the sequence )(m can be

expressed as

.|1|

|1|=)(

2

22

i

i

e

ef

By using (12) and (13) the function )( , 1> , is calculated by formula

.)(1)(11=)( 1)( iii eee

Thus increments of order 1 with step 0> of the sequence )(m form one-sided moving average

stochastic sequence of order .

Consider the problem of finding the mean square optimal linear estimate of the value of the

functional (1)(0)=1 baA which depends of unknown values (0) , (1) of the stochastic

sequence )(m from observations )(m at points 2,1,= m . We use theorem 3.3 to solve this

problem. The spectral characteristic (31) of the optimal estimate 1A of the functional 1A can be

calculated by the formula


137

,))(1(1

))(1)((1)(=)( 1

)(

,1

ii

iiia

ee

bebaebebah

where 1 is the Kronecker symbol. Using formula (30) we calculated an estimate of the functional

1A

).()())(1(=ˆ 1

1=

1 kbaA k

k

The value of mean square error is calculated by formula (33)

).2(2)(12=)ˆ,( 222

1 babaAf

4. Minimax-robust method of extrapolation The proposed formulas may be employed under the condition that the spectral density )(f of the

considered stochastic sequence )(m with stationary n th increments is known. The value of the

mean square error )ˆ,(:=));(( )( Afffh a and the spectral characteristic )()( fh a

of the

optimal linear estimate A of the functional A which depends of unknown values )(m can be

calculated by formulas (27) and (29) , the value of mean square error )ˆ,(:=));(( )(

, N

a

N Afffh

and the spectral characteristic )()(

, fh a

N of the optimal linear estimate NA of the functional NA

which depends of unknown values )(m can be calculated by formulas (31) and (33) . In the case

where the spectral density is not exactly known, but a set D of admissible spectral densities is given, the minimax (robust) approach to estimation of the functionals of the unknown values of a stochastic sequence with stationary increments is reasonable. In other words we are interesting in finding an estimate that minimizes the maximum of the mean square errors for all spectral densities from a given class D of admissible spectral densities simultaneously.

Definition 4.1 For a given class of spectral densities D a spectral density 0( )f D is called least

favorable in D for the optimal linear estimate the functional A if the following relation holds true:

( ) ( )

0 0 0( ) = ( ( ); ) = ( ( ); ).maxa a

f D

f h f f h f f

Definition 4.2 For a given class of spectral densities D a spectral characteristic )(0 h of the

optimal linear estimate of the functional A is called minimax-robust if there are satisfied conditions

0 0

2( ) = ( ),D

f D

h H L f

I


138

0( ; ) = ( ; ).max maxminh H f D f DD

h f h f

Analyzing the derived formulas and using the introduced definitions we can conclude that the following statements are true.

Lemma 4.1 Spectral density 0( )f D which admits the canonical factorization (12) is the least

favorable in the class of admissible spectral densities D for the optimal linear estimation of the functional A if

,)(=)(

2

0

0=

0 ki

k

ekf

(39)

where }0,1,2,=:)({= 00 kk is a solution to the conditional extremum problem

2

2

=0

|| || max, ( ) = ( ) .i k

k

D A f k e D

(40)

Lemma 4.2 Spectral density 0( )f D which admits the canonical factorization (12) is the least

favorable in the class of admissible spectral densities D for the optimal linear estimation of the

functional NA if

,)(=)(

2

0

0=

0 kiN

k

ekf (41)

where },0,1,2,=:)({= 00 NkkN is a solution to the conditional extremum problem

2

2

,

=0

|| || max, ( ) = ( ) .N

i k

N N N

k

D A f k e D

(42)

If ( ) 0( )a

Dh f H , the minimax-robust spectral characteristic can be calculated as )(= 0)(0 fhh a

.

The minimax-robust spectral characteristic 0h and the least favorable spectral density 0f form a

saddle point of the function );( fh on the set DH D . The saddle point inequalities

0 0 0 0( ; ) ( ; ) ( ; ) Dh f h f h f f D h H

hold true if )(= 0)(0 fhh a

and ( ) 0( )a

Dh f H , where 0f is a solution to the conditional extremum

problem


139

( ) 0( ) = ( ( ); ) inf , ,af h f f f D % (43)

,)()(

|)(|

2

1=));((

0

2

0)(

dff

erffh

i

a

where )(

ier is determined by formula (28) or (32) with )(=)( 0 ff .

The conditional extremum problem (43) is equivalent to the unconditional extremum problem

( ) = ( ) ( | ) inf ,f f f D %D

where ( | )f D is the indicator function of the set D . Solution 0f to this unconditional extremum

problem is characterized by the condition 00 ( )D f , where 0( )D f is the subdifferential of

the functional 0( )D f at point 0f (see Pshenichnyi (1982) or Moklyachuk (2008)). With the help

of the condition 00 ( )D f we can find the least favorable spectral densities in some special

classes of spectral densities (see books by Moklyachuk (2008), Moklyachuk and Masyutka (2012) for more details).

5. Least favorable spectral densities in the class 0D

Consider the problem of the optimal estimation of functionals A and NA of unknown values

)(k , 0,1,2=k , of the stochastic sequence )(k with stationary n th increments in the case

where the spectral density is not known, but the following set of spectral densities is given

0 0

1= ( ) | ( ) .

2f f d P

D

It follows from the condition 00 ( )D f for 0=D D that the least favorable density satisfies the

equation

,)(=))((|)(| 2102)( cfer ia

where 0)( and 0=)( if 0>)(0 f . Therefore, the least favorable density in the class 0D

for the optimal linear estimation of the functional A can be presented in the form

,)(=)(

2

0=

0 ki

k

k

eADcf

(44)

where the unknown parameters c , )(2),(1),(0),(= can be calculated using

factorization (12) , equation (37) , condition (40) and condition .2=)( 0Pdf

Consider the equation


140

= , .D AW C (45)

For each solution of this equation such that 0

2 =|||| P the following equality holds true:

.)(=)(=)(

2

0=

2

0=

0 ki

k

k

ki

k

eAWDcekf

Denote by 00P the maximum value of 2|||| AWD on the set of those solutions of equation

(45) , which satisfy condition 0

2 =|||| P and define canonical factorization (12) of the spectral

density )(0 f . Let 00 P be the maximum value of 2|||| AWD on the set of those which

satisfy condition 0

2 =|||| P and define canonical factorization (12) of the spectral density )(0 f

defined by (44) .

The derived equations and conditions give us a possibility to verify the validity of following statement.

Theorem 5.1 If there exists a solution 0}:)({= 00 mm of equation (45) which satisfies

conditions 0

20 =|||| P and 20

0000 ||=||= AWDPP , the spectral density (39) is least favorable

density in the class 0D for the optimal estimation of the functional A of unknown values )(k ,

0,1,2=k , of the stochastic sequence )(m with stationary n th increments. The increment

sequence ),()( mn admits a one-sided moving average representation. If

00 < , the density (44)

which admits the canonical factorization (12) is least favorable in the class 0D . The sequence

0}:)({= kkcc is determined by equality (37) , conditions (40) and the condition

02=)( Pdf

.

Consider the problem of optimal estimation of the functional NA . In this case the least favorable

spectral density is determined by the relation

.)(=)(

2

,

0=

0 ki

kNNN

N

k

eADcf

(46)

Define the matrix

ND with the help of relation

,,0,1,2,=),()()(=)ˆ(=0=

, NkNkldlmamADN

kNl

N

m

kNNN

(47)

where 0=)( pa if Np > . Taking into consideration (38) , we have the following equality


141

.)ˆ(=)(=)(

2

0=

2

0=

2)(

,

ji

jNNNN

N

j

ji

jNNNN

N

j

ia

N eWADeWADer (48)

Therefore each solution ))(,(2),(1),(0),(= 0000 NN of the equation

= , ,NN N N ND A W C (49)

or the equation

ˆ = , ,NN N N ND A W C (50)

such that 0

2 =|||| PN , satisfies the following equality

.)(=)(=)(2

)(

,

2

0=

0

ia

N

kiN

k

erekf (51)

Denote by 00 PN the maximum value of 22 ||ˆ=|||||| NNNNNNNN WADWAD on the set of solutions

N of equation (49) or equation (50) , which satisfy condition 0

2 =|||| PN and determine the

canonical factorization (12) of the spectral density 0

0( )f D . Let 00 PN be the maximum value

of 2|||| NNNN WAD

on the set of those N which satisfy condition 0

2 =|||| PN and determine the

canonical factorization (12 ) of the spectral density )(0 f defined by ( 46 ).


Theorem 5.2 If there exists a solution },0,1,2,=:)({= 00 NmmN of equation (49) or equation

(50) which satisfies conditions 0

20 =|||| PN and 20

0000 ||=||= NNNN WADPP , the spectral density

(41) is least favorable in the class 0D for the optimal estimation of the functional NA of unknown

values )(k , Nk ,0,1,2= , of the stochastic sequence )(m with stationary n th increments. The

increment sequence ),()( mn admits a one-sided moving average representation of order N . If

00 < , the density (46) which admits the canonical factorization (12) is least favorable in the

class 0D . The sequence },0,1,2,=:)({=, Nkkcc N is determined by equation (38) ,

conditions (42) and the condition 02=)( Pdf

.

Example 5.1 Consider the problem of minimax estimation of the functional (1)(0)=1 baA of

a stochastic sequence { ( ) : }m m Z with stationary increments of order 1 from observations of

the sequence )(m for 2,1,= m . We use theorem 5.2 to solve this problem. The matrices

used in (42) and (38) are the following


142

,(0)(1)

0(0)=,

10

1=,

0= 1

1

11

ww

wWD

b

baA

where (0)w , (1)w are the Fourier coefficients of the function )(

iew defined by (36) . The

least favorable density in the set 0D is defined by a solution of the optimization problem (42) ,

where 11,1, == DN , )(1)(0),(= 111

. Let us assume that 0xy , where

(1)(0))(:= 1 bwwbax , (0):= bwy . Then the optimization problem can be represented in

the form

,(1)(0)

;max(0)(1))(0)(

0

22

222

P

yyx

A solution )(1)(0),(= 000

1 of this problem is calculated as follows

;)42(

))4(4(=(0)

2

1

22

22222

00

yx

yxxyxP

.)42(

))4(4()(=(1)

2

1

22

22222

00

yx

yxxyxPxysign

The vector )(1)(0),(= 000

1 provides the maximum value of 2

1111 |||| WAD , satisfies condition

0

20

1 =|||| P and equation (49) with y

yxx

2

4=

22 if 0>y , and with

y

yxx

2

4=

22 if

0<y . Using theorem 5.2 we can conclude that the spectral density 2000 |(1)(0)=|)( ief

is the least favorable one in the class 0D for the optimal estimation of the functional

(1)(0)=1 baA of unknown values (0) , (1) of the stochastic sequence )(m with

stationary n th increments.

6. Least favorable spectral densities in the class M

D



where the spectral density is not exactly known, but the following set of spectral densities is given

,,0,1,2,=,=)(cos)(2

1|)(=

Mmdmff mM

D


143

where 00 = P and m{ , },0,1,2,= Mm is a strictly positive sequence (see Krein and Nudel'man

(1977)). It follows from the condition 00 ( )D f that the least favorable density satisfies the

equation

.cos)(=))((|)(|1=

102)(

mcfer m

M

m

ia

Thus, the least favorable density in the class MD for the optimal linear estimation of the functional

A can be presented in the form

,

)(

=)(2

1=

2

0=

0

0

ki

m

M

k

ki

k

k

ec

eADc

f

(52)

where parameters mc , Mm ,0,1,2,= , )(2),(1),(0),(= can be calculated using

conditions (40) , condition mdmf

2=)(cos)( , Mm ,0,1,2,= , equation (37) ,

factorization (12) .

Denote by 0PM the maximum value of 2|||| AWD on the set of solutions of the equation

(45) which satisfy condition 0

2 =|||| P and determine the canonical factorization (12 ) of the

spectral density )(0 f . Let 0PM

be the maximum value of 2|||| AWD on the set of those

which satisfy condition 0

2 =|||| P and determine the canonical factorization (12 ) of the spectral

density 0 ( ) Mf D defined by (52) . The derived equations and conditions give us a possibility to

verify the validity of following statement.


conditions 0

20 =|||| P and 20

000 ||=||= AWDPP M

, the spectral density (39) is least favorable

in the class MD for the optimal extrapolation of the functional A of unknown values )(k ,

0,1,2=k , of the stochastic sequence with stationary n th increments. If

MM < , the density

(52) which admits the canonical factorization (12) is least favorable in the class MD . The sequence

0}:)({= kk and unknown parameters mc , Mm ,0,1,2,= , are determined by equation

(37) , conditions (40) and conditions mdmf

2=)(cos)( , Mm ,0,1,2,= .

In the case of estimation of the functional NA the least favorable spectral density is defined by

equation


144

.

)(

=)(2

1=

2

,

0=

0

0

ki

m

M

k

ki

kNNN

N

k

ec

eADc

f

(53)

Let the matrix

ND be defined by equality (47) . Then equality (48) holds true. Therefore each

solution ))(,(2),(1),(0),(= 0000 NN of the equation (49) or the equation (50) such that

0

2 =|||| PN satisfies equality (51) .

Denote by 00 PN be the maximum value of 22 ||ˆ=|||||| NNNNNNNN WADWAD on the set of

solutions N of equation (49) or equation (50) , which satisfy condition 0

2 =|||| PN and

determine the canonical factorization (12) of the spectral density )(0 f . Let 00 PN be the

maximum value of 2|||| NNNN WAD

on the set of those N which satisfy condition 0

2 =|||| PN and

determine the canonical factorization (12) of the spectral density )(0 f defined by (53) .




20 =|||| PN and 20

0000 ||=||= NNNN WADPP , the spectral density

(41) is least favorable in the class MD for the optimal estimation of the functional NA of unknown

values )(k , Nk ,0,1,2= , of the stochastic sequence with stationary n th increments. If

00 < ,

the density (53) which admits the canonical factorization (12) is the least favorable in the class

MD . The unknown parameters },0,1,2,=:)({=, NkkN and mc , Mm ,0,1,2,= , are

determined by equality (38) , conditions (42) and conditions mdmf

2=)(cos)( ,

Mm ,0,1,2,= .

7. Least favorable spectral densities in the class u

vD




,)(2

1),()()(|)(= 0

Pdfufvfu

v

D


145

here )(v and )(u are some given (fixed) spectral densities. It follows from the condition

00 ( )D f for = u

vD D that the least favorable density 0f in the class u

vD for the optimal linear

estimation of the functional A is of the form

,)(),(min),(max=)(

2

0=

0

ki

k

k

eADcuvf

(54)

where the unknown parameters c , )(2),(1),(0),(= can be calculated using

factorization (12) , equation (37) , conditions (40) and condition .2=)( 0Pdf

Denote by 0Pu the maximum value of 2|||| AWD on the set of those solutions of equation

(45) , which satisfy inequalities

),()()(

2

0=

uekv ki

k

satisfy condition 0

2 =|||| P and determine the canonical factorization (12) of the spectral density

)(0 f . Let 0Pu

be the maximum value of 2|||| AWD on the set of those which satisfy

condition 0

2 =|||| P and determine the canonical factorization (12) of the spectral density )(0 f

defined by (54) . The derived equations and conditions give us a possibility to verify the validity of

the following statement.


conditions 0

20 =|||| P and 20

00 ||=||= AWDPP uu


in the class u

vD for the optimal estimation of the functional A of unknown values )(k , 0,1,2=k ,

of the stochastic sequence )(m with stationary n th increments. The increment sequence ),()( mn

admits one-sided moving average representation. If

uu < , the density (54) which admits the

canonical factorization (12) is least favorable in the class u

vD . The sequence 0}:)({= kkcc

is determined by equality (37) , conditions (40) and the condition 02=)( Pdf

. The minimax-

robust spectral characteristic is calculated by formulas (27) , (28) .

Consider the problem of the optimal estimation of the functional NA . In this case the least

favorable spectral density is determined by the relation

.)(),(min),(max=)(

2

0=

0

ki

k

N

k

eADcuvf

(55)


146

Denote by 0PN

u the maximum value of 22 ||ˆ=|||||| NNNNNNNN WADWAD on the set of solutions

N of equations (49) and (50) which satisfy inequality

),()()(

2

0=

uekv kiN

k

satisfy condition 0

2 =|||| PN and define the canonical factorization (12) of the spectral density 0 ( ) u

vf D . Let 0PN

u

be the maximum value of 2|||| NNNN WAD

on the set of those N which

satisfy condition 0

2 =|||| PN and define canonical factorization (12) of the spectral density )(0 f

determined by (55) .




20 =|||| PN and 20

00 ||=||= NNNN

N

u

N

u WADPP , spectral density

(41) is least favorable in the class u

vD for the optimal estimation of the functional NA of unknown

values )(k , Nk ,0,1,2= , of the stochastic sequence )(m with stationary n th increments. The

increment ),()( mn admits one-sided moving average representation of order N . If N

u

N

u < , the

density (55) which admits the canonical factorization (12) is least favorable in the class u

vD . The

sequence },0,1,2,=:)({=, Nkkcc N is determined by equation (38) , conditions (42) and

02=)( Pdf

. The minimax-robust spectral characteristic is calculated by formulas (31) , (32) .

Corollary 7.1 If we take 0=)(v and =)(u , two previous theorems give us solutions to the

problem of the minimax estimation of the functionals A and NA for the set of spectral densities

.)(2

1|)(= 00

PdffD

8. Least favorable spectral densities in the class

D





147

,|)()(|2

1|)(=

dvffD

where )(v is a bounded spectral density.

From the condition 00 ( )D f for =D D we find the following equation to determine the least

favorable spectral densities

.)(),(max=)(

2

0=

0

ki

k

k

eADcvf

(56)

Let us define

.=)(2

1=)(

2

11Pdvdf

(57)

Let 1P be the maximum value of 2|||| AWD on the set of those which belongs to the set of

solutions of equation (45) , satisfy the inequality

,)()(

2

0=

ki

k

ekv

satisfy condition 1

2 =|||| P and determine the canonical factorization (12) of the spectral density

)(0 f . Let 1P be the maximum value of 2|||| AWD on the set of those which satisfy

condition 1

2 =|||| P and determine the canonical factorization (12) of the spectral density )(0 f

defined by (56) . The following statement holds true.


conditions 1

20 =|||| P and 20

11 ||=||= AWDPP


in the class D for the optimal extrapolation of the functional A of unknown values )(k ,

0,1,2=k , of the stochastic sequence )(m with stationary n th increments. The increment

),()( mn admits one-sided moving average representation. If

uu < , the density (56) which

admits the canonical factorization (12) is least favorable in the class D . The sequence

0}:)({= kkcc is determined by equality (37) , conditions (40) and

dvdf )(2=)(

. The minimax-robust spectral characteristic is calculated by formulas

(27) , (28) .


148

In the case of optimal estimation of the functional NA the least favorable spectral density is

determined by formula

.)(),(max=)(

2

0=

0

ki

k

N

k

eADcvf

(58)

Let 1PN

be the maximum value of 22 ||ˆ=|||||| NNNNNNNN WADWAD on the set of those N

which belong to the set of solutions of equation (49) or equation (50) , satisfy the inequality

,)()(

2

0=

kiN

k

ekv

satisfy condition 1

2 =|||| PN and determined the canonical factorization (12) of the spectral

density )(0 f , 0( )f D . Let 1PN

be the maximum value of 2|||| NNNN WAD

on the set of

those N which satisfy condition 1

2 =|||| PN and determined the canonical factorization (12) of

the spectral density )(0 f defined by (58) . The following statement holds true.



20 =|||| PN and 20

11 ||=||= NNAWDPP

, the spectral density

)(41 is least favorable in the class D for the optimal extrapolation of the functional NA of

unknown values )(k , Nk ,0,1,2= , of the stochastic sequence )(m with stationary n th

increments. The increment ),()( mn admits one-sided moving average representation of order N .

If

< , the density (58) which admits the canonical factorization (12) is least favorable in the

class D . The sequence },0,1,2,=:)({=, Nkkcc N is determined by equation (38) ,

conditions (42) and

dvdf )(2=)(

. The minimax-robust spectral characteristic is

calculated by formulas (31) , (32) .

9. Conclusions In this article we describe methods of solution of the problem of optimal linear estimation of functionals which depend on unknown values of a stochastic sequence )(m with stationary n th

increments. Estimates are based on observations of the sequence )(t at points 2,1,= t .

Formulas are derived for computing the value of the mean-square error and the spectral characteristic of the optimal linear estimate of functionals in the case of spectral certainty where the spectral density of the sequence is exactly known.


149

In the case of spectral uncertainty where the spectral density is not exactly known but, instead, a set of admissible spectral densities is specified, the minimax-robust method is applied. We propose

a representation of the mean square error in the form of a linear functional in 1L with respect to

spectral densities, which allows us to solve the corresponding conditional extremum problem and describe the minimax (robust) estimates of the functional. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are derived for some concrete classes of admissible spectral densities.

References Dubovets'ka, I.I., Masyutka, O.Yu. and Moklyachuk, M.P., 2012. Interpolation of periodically correlated

stochastic sequences. Theory Probability and Mathematical Statistics 84, 43-156. http://dx.doi.org/10.1090/S0094-9000-2012-00862-4 Dubovets'ka, I.I., Moklyachuk, M.P., 2012. Filtering of periodically correlated processes. Prykladna Statystyka.

Aktuarna ta Finansova Matematyka 2, 149-158. Dubovets'ka, I.I., Moklyachuk, M.P., 2013. Extrapolation of periodically correlated processes from

observations with noise. Theory Probability and Mathematical Statistics 88, 60-75. Dubovets'ka, I.I., Moklyachuk, M.P., 2013. Minimax estimation problem for periodically correlated stochastic

processes. Journal of Mathematics and System Science 3(1), 26-30. Grenander, U., 1957. A prediction problem in game theory. Arkiv fuer Matematik 3, 371-379. http://dx.doi.org/10.1007/BF02589429 Franke, J., 1985. Minimax robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw.

Gebiete 68, 337-364. http://dx.doi.org/10.1007/BF00532645 Franke, J., Poor, H.V., 1984. Minimax-robust filtering and finite-length robust predictors, In: Robust and

Nonlinear Time Series Analysis. Lecture Notes in Statistics, Springer-Verlag 26, 87-126. Kassam, S.A., Poor, H.V., 1985. Robust techniques for signal processing: A survey. Proceedings of the IEEE 73,

433-481. http://dx.doi.org/10.1109/PROC.1985.13167 Karhunen, K., 1947. Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Annales Academiae

Scientiarum Fennicae. Series A I. Mathematica 37, 3-79. Kolmogorov, A.N., 1992. Selected works of A. N. Kolmogorov. Vol. II: Probability theory and mathematical

statistics. Ed. by A. N. Shiryayev. Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc.: Kluwer Academic Publishers, Moskva, 584.

Kurkin, O.M., Korobochkin, Yu. V. and Shatalov, S.A., 1990. Minimax information processing. Energoatomizdat, Moskva, 214.

Krein, M.G., Nudel'man, A.A., 1977. The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebysev and A. A. Markov and their further development. Translations of Mathematical Monographs. Vol. 50. Providence, R.I.: American Mathematical Society(AMS), Moskva, 552.

Luz, M. M., Moklyachuk, M. P., 2012. Interpolation of functionals of stochastic sequences with stationary increments. Theory Probability and Mathematical Statistics 87, 94-108.

Luz, M. M., Moklyachuk, M. P., 2012. Interpolation of functionals of stochastic sequences with stationary increments for observations with noise. Prykladna Statystyka. Aktuarna ta Finansova Matematyka 2, 131-148.

Moklyachuk, M. P., 1994. Stochastic autoregressive sequences and minimax interpolation. Theory Probability and Mathematical Statistics 48, 95-103.

Moklyachuk, M. P., 1998. Extrapolation of stationary sequences from observations with noise. Theory Probability and Mathematical Statistics 57, 133-141.

http://dx.doi.org/10.1090/S0094-9000-2012-00862-4

http://dx.doi.org/10.1007/BF02589429

http://dx.doi.org/10.1007/BF00532645

http://dx.doi.org/10.1109/PROC.1985.13167


150

Moklyachuk, M. P., 2000. Robust procedures in time series analysis. Theory Stochastic Processes 6(3-4), 127-147.

Moklyachuk, M. P., 2001. Game theory and convex optimization methods in robust estimation problems. Theory Stochastic Processes 7(1-2), 253-264.

Moklyachuk, M. P., 2008. Robust estimations of functionals of stochastic processes. Vydavnycho-Poligrafichnyi Tsentr, Kyivskyi Universytet, Kyiv, 320.

Moklyachuk, M.P., Masyutka, O.Yu., 2006. Interpolation of multidimensional stationary sequences. Theory Probability and Mathematical Statistics 73, 125-133.

http://dx.doi.org/10.1090/S0094-9000-07-00687-4 Moklyachuk, M.P., Masyutka, O.Yu., 2006. Extrapolation of multidimensional stationary processes. Random

Operators Stochastic Equations 14(3), 233-244. http://dx.doi.org/10.1515/156939706778239819 Moklyachuk, M.P., Masyutka, O.Yu., 2007. On the problem of filtration of vector stationary sequences. Theory

Probability and Mathematical Statistics 75, 109-119. http://dx.doi.org/10.1090/S0094-9000-08-00718-7 Moklyachuk, M.P., Masyutka, O.Yu., 2008. Minimax prediction problem for multidimensional stationary

stochastic sequences. Theory Stochastic Processes 14(3-4),89-103. Moklyachuk, M.P., Masyutka, O.Yu., 2011. Minimax prediction problem for multidimensional stationary

stochastic processes. Communications in Statistics. Theory and Methods 40, 3700-3710. http://dx.doi.org/10.1080/03610926.2011.581190 Moklyachuk, M. P., Masyutka, O.Yu., 2012. Minimax-robust estimation technique for stationary stochastic

processes. LAP LAMBERT Academic Publishing, 296. Pinsker, M. S., Yaglom, A. M., 1954. On linear extrapolation of random processes with nth stationary

increments. Doklady Akademii Nauk SSSR 94, 385—388. Pinsker, M. S., 1955. The theory of curves with nth stationary increments in Hilbert spaces. Izvestiya

Akademii Nauk SSSR. Ser. Mat. 19(5), 319-344. Pshenichnyi, B.N., 1982. Necessary conditions for an extremum. 2nd ed., Nauka, Moskva, 144. Rozanov, Yu. A., 1990. Stationary stochastic processes. 2nd rev. ed. Nauka, Moskva, 272. Vastola, K. S., Poor, H. V., 1983. An analysis of the effects of spectral uncertainty on Wiener filtering.

Automatica 28, 289-293. http://dx.doi.org/10.1016/0005-1098(83)90105-X Wiener, N., 1966. Extrapolation, interpolation and smoothing of stationary time series. With engineering

applications. The M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 163. Yaglom, A. M., 1987. Correlation theory of stationary and related random functions. Vol. 1: Basic results.

Springer Series in Statistics, Springer-Verlag, New York etc., 526. Yaglom, A. M., 1987. Correlation theory of stationary and related random functions. Vol. 2: Supplementary

notes and references. Springer Series in Statistics, Springer-Verlag, New York etc., 258. Yaglom, A. M., 1955. Correlation theory of stationary and related random processes with stationary nth

increments. Mat. Sbornik 37(1), 141-196. Yaglom, A. M., 1957. Some classes of random fields in n-dimensional space related with random stationary

processes. Teor. Veroyatn. Primen. 2, 292-338.

http://dx.doi.org/10.1090/S0094-9000-07-00687-4

http://dx.doi.org/10.1515/156939706778239819

http://dx.doi.org/10.1090/S0094-9000-08-00718-7

http://dx.doi.org/10.1080/03610926.2011.581190

http://dx.doi.org/10.1016/0005-1098(83)90105-X

robust extrapolation problem for stochastic sequences · pdf filethe mean square optimal...

Documents