tikhonov’s regularization to the deconvolution problem
TRANSCRIPT
TIKHONOV’S REGULARIZATION TO
DECONVOLUTION PROBLEM
Dang Duc Tronga, Dinh Ngoc Thanha, Truong Trung Tuyenb, and Cao Xuan Phuongc
a Department of Mathematics and Computer Science, HoChiMinh City National University, 227
Nguyen Van Cu, HoChiMinh City, VietNam.
b Department of Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405, USA.
c Department of Mathematics and Statistics, Ton Duc Thang University, District 7, HoChiMinhCity, VietNam.
Abstract
We are interested in estimating the pdf f of the i.i.d. random variables X1, . . . , Xn from themodel Yi = Xi + Zi, where Zi are unobserved error random variables, distribute with the densityfunction g and independent of Xi. This problem is known as deconvolution problem in nonparametricstatistic. The most popular method of solving of the problem is the kernel one in which, we assumegft(t) 6= 0, for all t ∈ R, where gft(t) is the Fourier transform of g. The case in which gft(t) has realzeros do not consider much. In this paper, we shall consider this case. In fact, by estimating the setof small-valued of gft and by combining with the Tikhonov’s regularization method, we introduce anapproximation fn to the density function f and evaluate the rate of the convergent E ‖fn − f‖2
L2(R).
AMS subject classifications: 62G07, 62F05.
Key words: Deconvolution, Tikhonov’s regularization, Cartan’s theorem, Fourier transform.
1 Introduction
In this paper, we are interested in estimating the density function f of the random vari-
ables i.i.d X1, X2, . . . , Xn base on the direct random variables Y1, Y2, . . . , Yn from model
Yi = Xi + Zi, i = 1, 2, . . . , n. (1)
Here Zi are unobserved error random variables, distribute with the density function g and
independent of Xi. We known that if h is the probability density function of Yi, we have
relative
h (x) = (f ∗ g) (x) :=
∫ +∞
−∞f (x− t) g (t) dt, x ∈ R. (2)
where symbol ∗ denotes the convolution of f and g.
We denote the Fourier transform of the function f by
f ft (t) =
∫ +∞
−∞f (x) eitxdx. (3)
1
Informally, if h is known, we can apply the Fourier transform to both sides of (2) to
get
f ft =hft
gft. (4)
Then using the inverse Fourier tranform, we can find f . This is a classical problem in
Analysis.
In practical situations, we do not have the density h. We only have the observations
Yi, i = 1, . . . , n. The problem of recovering f from observations Yi distributed according
to h is called the deconvolution problem in statistics or deconvolution problem for short.
Equation (2) is an intergral equation and solving (2) is typically an ill-posed problem.
Deconvolution is known to be difficullty. The most popular approach to deconvolution is
the use of a kernel estimator of f obtained by applying the Fourier inversion formular to
the empirical characteristic function of X. They approximate the density function f by
the function
f̂ (x;Y1, . . . , Yn) =1
2π
∫ +∞
−∞e−itxK
ft (tb)
gft (t)
1
n
n∑
j=1
eitYjdt, (5)
where K is the kernel function, Kft has compactly supported, and gft(t) 6= 0 for all t ∈ R.
This method was initied by the papers of Carrol and Hall [4], Stefanski and Carrol [5],
later followed by Fan [6], [7], [8]. The condition
gft (t) 6= 0, ∀t ∈ R
has become common in deconvolution topics although there are important densities that
do not satisfy, for instance the uniform density g on the interval [−1; 1] having the Fourier
transform gft (t) = (sin t) /t.
Recently, Hall and Meister [9] have given a approach to deconvolution problem in
case of the Fourier transform of error distribution has zeros. In case gft (t) has real zeros,
to estimate for density function f , authors have used the function
f̂ (x) = Re
1
2π
∫ +∞
−∞e−itx gft (−t)
∣∣f ft (t)∣∣r
(max
{∣∣gft (t)∣∣ ;n−ξ|t|ρ
})r+2
1
n
n∑
j=1
eitYjdt
, (6)
with r ≥ 0, ξ > 0 and ρ ≥ 0. In case g in gνµλ the class of probability densities g satisfy
gft(t) does not vanish for |t| ≤ T and
C1|sin (λt)|µ|t|−v ≤∣∣∣gft (t)
∣∣∣ ≤ C2|sin (λt)|µ|t|−v, |t| > T, (7)
where µ ≥ 1, ν > 0, 0 ≤ C1 ≤ C2, λ > 0 and T > 0, the rate of the convergent of mean
square error is polynomial rates. In case g ∈ g′νµλ, the class of all densities g satisfying (7)
when |t|−ν is replaced by exp(−d|t|γ
), with d, γ > 0, the rate of the convergent of mean
2
square error is logarithmic rates. However, condition (7) imposed on gft is not natural.
In indeed, on R\ [−T ;T ], gft (t) = 0 if and only if t = kπλ for k ∈ Z. In other words,
they require the positions which gft vanish. If g is the uniform density on [−1; 1] then it
is obviously to see g satisfy (7). If g is an arbitrary density function then g is not sure
satisfy (7).
Motivated by this problem, in the present paper, we shall consider the deconvolution
problem in the case Fourier tranform of error distribution have zeros on the real line. Using
properties of entire functions, we consider small-value sets of error distribution. Applying
Tikhonov’s regularization, we introduce an estimation procedure for the density function
and evaluate the speed of convergence.
The rest of our paper consists of three sections. In Section 2, we will present the
Tikhonov’s regularization; base on Tikhonov’s regularization, we give a estimation for
probability density function. In Section 3, we state and prove approximation results.
In Section 4, by estimating of the size of small-valued sets Fourier transform, we prove
Lemma 3.1 which stated without proved in the previous sections.
2 Tikhonov’s regularization
As disscussed in Section 1, we have known that (2) is typically an ill-posed problem and
a regularization is required. In Theory of ill-posed problems, a method regularization
which often used to deconvolution problem is Tikhonov’s regularization. In this method,
we shall approximate f ft by a function having the form ϕgft where ϕ is often called
the ”filter function”. In fact, we consider the linear operator A : L2 (R) → L2 (R),
A (ϕ) = ϕgft for all ϕ ∈ L2(R). For each δ > 0, we consider Tikhonov’s functional
Jδ (ϕ) =∥∥∥Aϕ− hft
∥∥∥2
L2(R)+ δ ‖ϕ‖2
L2(R) , ϕ ∈ L2(R). (8)
We shall find the function ϕ minimizing Jδ. As known, Jδ attains its minimum at a unique
minimum function ϕδ ∈ L2(R). This minimum ϕδ is the unique solution of the equation
δϕδ + (A∗A) (ϕδ) = A∗(hft), (9)
with A∗ : L2(R) → L2(R) is the adjoint of A.
From (9), we get
δϕδ +∣∣∣gft∣∣∣2ϕδ = gft hft
and so we have the aproximation
ϕδ =gft
δ +∣∣gft∣∣2
3
to the Fourier transform f ft of the density function f . After that, using the inverse Fourier
transform gives
fδ (x) =1
2π
∫ +∞
−∞e−itx g
ft (t)hft (t)
δ +∣∣gft (t)
∣∣2dt, x ∈ R (10)
It can be seen as an estimation for density function f .
In practical situations, we do not have density h, we have only the observations
Y1, . . . , Yn. Thus, we cannot use directly the formula (10) to make an approximation for
f . However, in case we have the observations i.i.d Y1, . . . , Yn, since
E
1
n
n∑
j=1
eitYj
=1
n
n∑
j=1
E
(eitYj
)= hft (t) ,
we can replace hft (t) in (10) by the quantity
ψ (t;Y1, ..., Yn) =1
n
n∑
j=1
eitYj . (11)
It implies an approximation for density function f based on Y1, . . . , Yn as follows
Lδ(x;Y1, . . . , Yn) =1
2π
∫ +∞
−∞e−itx gft (t)
δ +∣∣gft (t)
∣∣21
n
n∑
j=1
eitYjdt. (12)
We have the following general estimate for error E ‖Lδ − f‖2L2(R).
Theorem 2.1 Let ε > 0, δ > 0, g ∈ L1 (R) ∩ L2 (R) is the density function of the error
random variables and f ∈ L1 (R) ∩ L2 (R) be the solution of Problem (2). Let β ∈ (0, 1)
and (Rε) be a sequence such that
limε→0
Rε = +∞.
Then there exists a constant C1 independent of ε, δ such that
E ‖Lδ − f‖2L2(R) ≤ C1
(m (Bεβ) +
∫
|t|>Rε, |gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2dt+
δ2
ε4β+
1
nδ2
), (13)
where
Bεβ ={t ∈ R :
∣∣∣gft (t)∣∣∣ < εβ , |t| < Rε
},
and m (Bεβ) denotes Lebesgue’s measure of set Bεβ .
Proof. From (12), we get
Lftδ (t) =
gft (t)
δ +∣∣gft (t)
∣∣21
n
n∑
j=1
eitYj , t ∈ R.
4
Applying the Parseval’s equality and the Fubini’s theorem, we derive
E ‖Lδ − f‖2L2(R) =
1
2π
∫ +∞
−∞E
∣∣∣Lftδ (t) − f ft (t)
∣∣∣2dt.
We have
E
∣∣∣Lftδ (t)− f ft (t)
∣∣∣2
= VarLftδ (t) +
∣∣∣ELftδ (t)− f ft (t)
∣∣∣2,
and so
E ‖Lδ − f‖2L2(R) =
1
2π
∫ +∞
−∞VarLft
δ (t) dt+1
2π
∫ +∞
−∞
∣∣∣ELftδ (t) − f ft (t)
∣∣∣2dt.
At first, since 0 ≤∣∣hft (t)
∣∣ ≤ 1, we have the following estimate
∫ +∞
−∞VarLft
δ (t)dt =
∫ +∞
−∞
1
n
∣∣∣∣∣gft (t)
δ +∣∣gft (t)
∣∣2
∣∣∣∣∣
2
Var(eitY1
)dt ≤
1
nδ2
∫ +∞
−∞
∣∣∣gft (t)∣∣∣2dt. (14)
In addition, since from the independence of Y1, Y2, . . . , Yn, we have
ELftδ (t) =
gft (t)
δ +∣∣gft (t)
∣∣2hft (t) =
∣∣gft (t)∣∣2
δ +∣∣gft (t)
∣∣2f ft (t) .
It follows that
∫ +∞
−∞
∣∣∣ELftδ (t) − f ft (t)
∣∣∣2dt =
∫ +∞
−∞
∣∣∣f ft (t)∣∣∣2(
δ
δ +∣∣gft (t)
∣∣2
)2
dt. (15)
Now we write
∫ +∞
−∞
∣∣∣f ft (t)∣∣∣2(
δ
δ +∣∣gft (t)
∣∣2
)2
dt =
∫
|t|<Rε, |gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2(
δ
δ +∣∣gft (t)
∣∣2
)2
dt
+
∫
|t|>Rε, |gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2(
δ
δ +∣∣gft (t)
∣∣2
)2
dt
+
∫
|gft(t)|>εβ
∣∣∣f ft (t)∣∣∣2(
δ
δ +∣∣gft (t)
∣∣2
)2
dt
≤
∫
|t|<Rε, |gft(t)|<εβ
dt+
∫
|t|>Rε, |gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2dt+
(δ
ε2β
)2 ∫
|gft(t)|>εβ
∣∣∣f ft (t)∣∣∣2dt.
Since∫
|gft(t)|>εβ
∣∣∣f ft (t)∣∣∣2dt ≤
∫ +∞
−∞
∣∣∣f ft (t)∣∣∣2dt =
∥∥∥f ft∥∥∥
2
L2(R)= 2π ‖f‖
2L2(R) ,
∫
|t|>Rε, |gft(t)|<εβ
dt = m (Bεβ) ,
5
we have
∫ +∞
−∞
∣∣∣ELftδ (t) − f ft (t)
∣∣∣2dt =
∫ +∞
−∞
∣∣∣f ft (t)∣∣∣2(
δ
δ +∣∣gft (t)
∣∣2
)2
dt
≤ m (Bεβ) +
∫
|t|>Rε, |gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2dt+ 2π ‖f‖2
L2(R)
δ2
ε4β.
The latter inequality implies
E ‖Lδ − f‖2L2(R) ≤ C1
(m (Bεβ) +
∫
|t|>Rε, |gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2dt+
δ2
ε4β+
1
nδ2
),
where C1 = max(1; 2π ‖f‖
2L2(R)
). The proof of theorem has completed.
3 Approximation results
The difficulty of applying directly the formula (4) arises in two aspects: one, in reality we
can not have the exact data h, and two, even we have h, we cannot compute efficiently
the inverse Fourier tranform of f ft if the function gft has zeros on the real axis. Hence, a
regularization is in order.
For each entire function ψ, we say that {z ∈ C : |ψ (z)| < ε} is the small-valued set
of ψ. In case g has compact support, the set of zeros gft(t) effects risky to the recovering
the function f from its Fourier tranform. For really computing the solution f and for
the regularization of equation (2), we must known more about the Lebesgue measure
of the small-valued set of gft. This problem goes back to the well-known theorem of
Cartan about the size of the small-valued sets Aε = {z ∈ C : |P (z)| < ε} where P (z) is
a polynomial. He proved that Aε is contained in a finite of disks whose sum of radius is
less than Cε1n , where n is the degree of P (z) and C is a constant that depends only on
the leading coefficient of P (z) and n (see Theorem 3 of §11.2 in [3]). In particularly, we
have
limε→0
m ({z ∈ C : |P (z)| < ε}) = 0
where m(.) is the Lebesgue measure.
In this paper, we shall use Cartan’s theorem to give some asymptotic estimates for
the small-valued set {t ∈ R :
∣∣∣gft (t)∣∣∣ < ε, |t| < R
}
of the function g and to apply this estimate to the Tikhonov’s regularization of problem
(2).
6
For each ε > 0, we put
sε = inf
{s > 0 :
∫
|t|≥s
|g (t)| dt ≤ ε
}. (16)
Lemma 3.1 Let g is the density function of the error distribution, β ∈ (0, 1), q ∈ N. For
ε > 0 small enough, choose Rε satisfy
2esεRε
[(q +
1
2
)lnRε + ln
(15e3
)]= − ln
(εβ + ε
). (17)
Then
m (Bεβ) ≤ R−q+ 1
2ε ,
where
Bεβ ={t ∈ R :
∣∣∣gft (t)∣∣∣ < εβ , |t| < Rε
}.
We shall prove Lemma 3.1 in Section 4.
To get an explicit estimate for E ‖Lδ − f‖2L2(R), some a prior information about g
and f must be assumed. From now on, we assume f satisfies∣∣∣f ft (t)
∣∣∣2≤
C2(1 + |t|2
)q , q ∈ N. (18)
The condition (18) imposed on f is quite natural. It is equivalent to the condition that f
is in the Sobolev space Hq(R).
Moreover, let s0 > 0 and γ > 1, we assume that the density function g of the error
distribution satisfies ∫ +∞
−∞|g (t)| es0|t|
γ
dt < +∞. (19)
We note that the density functions g which has a compact support satisfy the latter
condition. For these functions, we do not sure that gft(t) 6= 0 for every t ∈ R. Our main
result is
Theorem 3.2 Let α ∈ (0, 1), β ∈ (0, 1), αβ < 14 and ν = 1
4 +αβ. Assume that f satisfies
(18). By choosing
ε = n−α, δ = n−ν, (20)
and denoting
fn (x) = Lδ (x;Y1, . . . , Yn) ,
we have the following estimate
E ‖fn − f‖2L2(R) ≤ C3
(
(s0)1γ
30 (2q + 1) e4
)− q
2+ 1
2
(ln (nα))(12−
12γ )(−q+ 1
2 ) + 2(ln (nα))2ν−1
α
,
where C3 is a constant depended on f .
7
Proof. Because f satisfies (18), we have∫
|t|>Rε,|gft(t)|<εβ
∣∣∣f ft (t)∣∣∣2dt ≤
∫
|t|>Rε
C2dt
(1 + t2)q ≤
∫
|t|>Rε
C2dt
|t|2q≤
2C2
2q − 1R−q+ 1
2ε
for all ε is small enough. Hence, combined with Theorem 2.1 and Lemma 3.1, we get
E ‖Lδ − f‖2L2(R) ≤ C1
[(1 +
2C2
2q − 1
)R−q+ 1
2ε +
δ2
ε4β+
1
nδ2
]
≤ C3
(R−q+ 1
2ε +
δ2
ε4β+
1
nδ2
)
for all ε is small enough, where C3 = C1
(1 + 2C2
2q−1
).
From (26), for all ε small enough, we have estimate
R−q+ 1
2ε ≤
((s0)
1γ
30 (2q + 1) e4
)− q
2+ 1
2(ln
(1
ε
))( 12− 1
2γ )(−q+ 12).
Therefore
E ‖Lδ − f‖2L2(R) ≤ C3
(
(s0)1γ
30 (2q + 1) e4
)− q
2+ 12(
ln
(1
ε
))( 12−
12γ )(−q+ 1
2)+
δ2
ε4β+
1
nδ2
.
Replacing ε = n−α, δ = n−ν , we get
E ‖fn − f‖2L2(R) ≤ C3
(
(s0)1γ
30 (2q + 1) e4
)− q
2+ 12
(ln (nα))(12−
12γ )(−q+ 1
2) + 2n2ν−1
for all n small enough. Furthermore, we have
n2ν−1 = (nα)2ν−1
α ≤ (ln (nα))2ν−1
α .
Hence
E ‖fn − f‖2L2(R) ≤ C3
(
(s0)1γ
30 (2q + 1) e4
)− q
2+ 12
(ln (nα))(12−
12γ )(−q+ 1
2 ) + 2(ln (nα))2ν−1
α
for all n small enough. The proof of theorem has completed.
If g has compact support then we get the following result
Theorem 3.3 Let assumptions be as in Theorem 3.2. Moreover, assume the density
function g has compact support, supp g ⊂ [−M ;M ], where supp g is the support of g.
Then
E ‖fn − f‖2L2(R) ≤ C3
[(1
30M (2q + 1) e4
)− q
2+ 1
4
(ln (nα))−q
2+ 1
4 + 2(ln (nα))2ν−1
α
],
where C3 is a constant depended on f .
8
Proof. For each ε > 0, from (16), we get∫|t|≥sε
|g (t)| dt ≤ ε. Moreover, from the property
of infimum, we have∫|t|≥sε−η |g (t)| dt > ε for all η > 0, this implies
∫|t|≥sε
|g (t)| dt ≥ ε.
So ∫
|t|≥sε
|g (t)| dt = ε.
If sε > M then ∫
|t|≥sε
|g (t)| dt = 0,
which is a contradiction. So sε ≤ M for all ε small enough.
From (22), for all ε small enough, we have
R2ε ≥
1
15 (2q + 1) e4
[1
Mβln
(1
ε
)+
1
Mln
(1
1 + ε1−β
)]≥
1
30M (2q + 1) e4ln
(1
ε
).
It follows that
R−q+ 1
2ε ≤
(1
30M (2q + 1) e4
)− q
2+ 14(
ln
(1
ε
))− q
2+ 14
.
Therefore, combined with the proof of Theorem 3.2, we derive
E ‖Lδ − f‖2L2(R) ≤ C3
[(1
30M (2q + 1) e4
)− q
2+ 1
4(
ln
(1
ε
))− q
4+ 1
4
+δ2
ε4β+
1
nδ2
].
Replacing ε = n−α, δ = n−ν , we get
E ‖fn − f‖2L2(R) ≤ C3
[(1
30M (2q + 1) e4
)− q
2+ 1
4
(ln (nα))−q
2+ 14 + 2(ln (nα))
2ν−1α
].
4 Proof of Lemma 3.1
To estimate the measure of the small-valued sets, we shall use the following result (see
Theorem 4, Section §11.3 in [3]).
Lemma 4.1 Let f(z) be a function analytic in the disk {z : |z| ≤ 2eR}, |f(0)| = 1, and
let η be an arbitrary small positive number. Then the estimate
ln |f (z)| > − ln
(15e3
η
). lnMf (2eR)
is valid everywhere in the disk {z : |z| ≤ R} except a set of disks (Cj) with sum of radius∑rj ≤ ηR, where Mf (r) = max
|z|=r|f (z)| .
9
Theorem 4.2 Let the density function g of error distribution satisfies (19), β ∈ (0, 1)
and q ∈ N. For ε > 0 small enough, we choose sε as in (16) and Rε to satisfy
2esεRε
[(q +
1
2
)lnRε + ln
(15e3
)]= − ln
(εβ + ε
). (21)
Then
limε→0
Rε = +∞,
and if ε small enough, we have
m (Bεβ+ε) ≤ R−q+ 1
2ε ,
where
Bεβ+ε ={z ∈ C : |Φε (z)| < εβ + ε, |z| < Rε
}
with
Φε (z) =
∫ sε
−sε
g (t) eztdt, z ∈ C.
Proof. Consider the function
ψ (R) = 2esεR
[(m+
1
2
)lnR + ln
(15e3
)]+ ln
(εβ + ε
), R ≥ 0.
We have ψ (R) → +∞ as R → +∞ and ψ(R) → ln(εβ + ε
)< 0 as R → 0 for ε small
enough. So there exists Rε > 0 such that ψ(Rε) = 0, i.e, Rε satisfies (21).
From (21), we get
(2q + 1) eRε ln(15e3Rε
)≥
ln(
1εβ+ε
)
sε,
and so
R2ε ≥
1
15e4 (2q + 1)
1β ln
(1ε
)+ ln
(1
1+ε1−β
)
sε. (22)
From (16), we get ∫
|t|≥sε
|g (t)| dt = ε.
Thus
e−s0(sε)γ
∫ +∞
−∞es0|t|
γ
|g (t)| dt ≥ e−s0(sε)γ
∫
|t|≥sε
es0|t|γ
|g (t)| dt ≥ ε.
This implies
sε ≤
(1
s0
) 1γ(
ln
(C4
ε
)) 1γ
, (23)
where C4 =∫ +∞−∞ es0|t|
γ
|g (t)| dt ≥ 1.
10
From (22) and (23), we get
R2ε ≥
1
15e4 (2q + 1)
(s0)1γ
β
ln(
1ε
)
(ln(
C4
ε
)) 1γ
+ (s0)1γ .
ln(
11+ε1−β
)
(ln(
C4
ε
)) 1γ
(24)
Because
limε→0
ln(
1ε
)
(ln(
C4
ε
)) 1γ
:
(ln
(C4
ε
)) 12− 1
2γ
= +∞,
we get
ln(
1ε
)
(ln(
C4
ε
)) 1γ
≥
(ln
(C4
ε
))12− 1
2γ
, (25)
for all ε small enough.
Furthermore, since
limε→0
ln(
11+ε1−β
)
(ln(
C4
ε
)) 1γ
= 0,
we have
R2ε ≥
(s0)1γ
30 (2q + 1)βe4
(ln
(C4
ε
)) 12−
12γ
≥(s0)
1γ
30 (2q + 1) e4
(ln
(1
ε
)) 12−
12γ
(26)
for all ε small enough. This follows that Rε → +∞ as ε → 0.
We define
gε (t) =
{g (t) , |t| ≤ sε,
0, |t| > sε.(27)
Since ‖g‖L1(R) = 1, there exists x0 ∈ R such that gft (x0) 6= 0. Then there exists a constant
C5 > 0 such that∣∣gft
ε (x0)∣∣ ≥ C5 if ε small enough. Changing variable if necessary, we
may assume that |Φε (0)| ≥ C5 if ε small enough.
For all |z| = 2eRε, we have
|Φε (z)| =
∣∣∣∣∫ sε
−sε
g (t) eztdt
∣∣∣∣ ≤∫ sε
−sε
|g (t)| e2esεRεdt ≤ e2esεRε,
It follows that
lnMΦε(2eRε) ≡ ln
(max
|z|=2eRε
|Φε (z)|
)≤ 2esεRε.
For all |z| ≤ Rε, we choose
ηε = R−q− 1
2ε
11
and apply Lemma 4.1, we have the estimate
|Φε (z)| ≥ exp
{− ln
(15e3
R−q− 1
2ε
)lnMΦε
(2eRε)
}
≥ exp
{−2esεRε
[(q +
1
2
)lnRε + ln
(15e3
)]}
= εβ + ε
except a set of disks whose sum of radius is less than ηεRε = R−q+ 1
2ε .
Proof of Lemma 3.1
For all x ∈ R, we have
∣∣∣gftε (x)− gft (x)
∣∣∣ =∣∣∣∣∫ sε
−sε
g (t) eitxdt−
∫ +∞
−∞g (t) eitxdt
∣∣∣∣ ≤∫
|t|≥sε
|g (t)| dt = ε.
Thus, if∣∣gft (x)
∣∣ < εβ , |x| < Rε then∣∣gft
ε (x)∣∣ < εβ + ε. This implies |Φε (ix)| < εβ + ε.
Applying Theorem 4.2, we get
m (Bεβ) ≤ m (Bεβ+ε) ≤ R−q+ 1
2ε .
References
[1] Alexander Meister, Deconvolution Problems in Nonparametric Statistics, Springer,
2009.
[2] Andreas Kirsch, An Introduction to the Mathematical Theory of Inverse Problems,
Springer- Verlag, New York, 1996.
[3] B. Ya. Levin, Lectures on Entire Functions, Trans. Math. Monographs, Vol. 150,
AMS, Providence, Rhole Island, 1996.
[4] Raymond Carroll and Peter Hall, Optimal Rates of Convergence for Deconvolving a
Density, Journal of American Statistical Association, (1988) Vol. 83, No. 404, 1184 -
1186.
[5] Stefanski, L. and R. Carroll, Deconvoluting Kernel Density Estimators, Statistics,
(1990) 2, 169 - 184.
[6] JianQing Fan, On the optimal rates of convergence for nonparametric deconvolution
problems, The Annals of Statistics, (1991) Vol. 19, No. 3, 1257 - 1272.
[7] JianQing Fan, Asymptotic normality for deconvolution kernel density estimators,
Sankhya, (1991) 53, 97 - 110.
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[8] JianQing Fan, Deconvolution with supersmooth distributions, The Canadian Journal
of Statistics, (1992) 20, 155 - 169
[9] Peter Hall and Alexander Meister, A ridge- parameter approach to deconvolution,
The Annals of Statistics, (2007) Vol. 35, No. 4, 1535 - 1558.
Dang Duc Trong, Department of Mathematics and Computer Science, HoChiMinh
City National University, 227 Nguyen Van Cu, HoChiMinh City, VietNam.
Email address : [email protected]
Dinh Ngoc Thanh, Department of Mathematics and Computer Sciences, HoChiMinh City
National University, 227 Nguyen Van Cu, HoChiMinh City, VietNam.
Email address : [email protected]
Truong Trung Tuyen, Department of Mathematics, Indiana University, Rawles Hall,
Bloomington, IN 47405, USA.
Email address : [email protected]
Cao Xuan Phuong, Department of Mathematics and Statistics, Ton Duc Thang University,
District 7, HoChiMinh City, VietNam.
Email address : [email protected]
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