Estimation of a fold convolution in additive noise model with compactly supported noise density
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76
SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018
Estimation of a fold convolution in
additive noise model with compactly
supported noise density
Cao Xuan Phuong
Abstract – Consider the model Y X Z ,
where Y is an observable random variable, X
is an unobservable random variable with
unknown density f , and Z is a random noise
independent of X . The density g of Z is
known exactly and assumed to be compactly
supported. We are interested in estimating the
m - fold convolution f m f f on the basis
of independent and identically distributed
(i.i.d.) observations Y1 , , Yn drawn from the
distribution of Y . Based on the observations as
well as the ridge-parameter regularization
method, we propose an estimator for the
function f m depending on two regularization
parameters in which a parameter is given and a
parameter must be chosen. The proposed
estimator is shown to be consistent with respect
to the mean integrated squared error under
some conditions of the parameters. After that
we derive a convergence rate of the estimator
under some additional regular assumptions for
the density f .
Index Terms – estimator, compactly supported
noise density, convergence rate
1 INTRODUCTION
I
n this paper, we consider the additive noise
model
Y X Z
(1)
where Y is an observable random variable, X is an
unobservable random variable with unknown
density f , and Z is an unobservable random noise
with known density g . The density g is called
noise density. We also suppose that X and Z are
independent. Estimating f on basis of i.i.d.
Received 06-05-2017; Accepted 15-05-2017; Published 108-2018
Author: Cao Xuan Phuong- Ton Duc Thang University ()
observations of Y has been known as the density
deconvolution problem in statistics. This problem
has received much attention during two last
decades. Various estimation techniques for f can
be found in Carroll-Hall [1], Stefanski-Carroll [2],
Fan [3], Neumann [4], Pensky-Vidakovic [5],
Hall-Meister [6], Butucea-Tsybakov [7], Johannes
[8], among others.
This problem has concerned with many real-life
problems in econometrics, biometrics, signal
reconstruction, etc. For example, when an input
signal passes through a filter, output signal is
usually disturbed by an additional noise, in which
the output signal is observable, but the input signal
is not.
Let Y1 , , Yn be n i.i.d. observations of Y .
In the present paper, instead of estimating f ,
we focus on the problem of estimating the m -fold
convolution
fm f f , m ,
(2)
m times
based on the observations. In the free-error case,
i.e. Z 0 , there are many papers related to this
problem, such as Frees [9], Saavedra-Cao [10],
Ahmad-Fan [11], Ahmad-Mugdadi [12], Chesneau
et al. [13], Chesneau-Navarro [14], and references
therein. For m 1 , the problem of estimating f m
reduces to the density deconvolution problem. To
the best of our knowledge, for m , m 2 , so
far this problem has been only studied by
Chesneau et al. [15]. In that paper, the authors
constructed a kernel type of estimator for f m under
the assumption that g ft is nonvanishing on ,
where the function g ft t f x eitx dt is the
Fourier transform of g . The latter assumption is
fulfilled with many usual densities, such as
normal, Cauchy, Laplace, gamma, chi-square
densities. However, there are also several cases of
g that cannot be applied to this paper. For
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instance, the case in which g is a uniform density
or a compactly supported density in general. In the
present paper, as a continuation of the paper of
Chesneau et al. [15], we consider the case of
compactly supported noise density g . In fact, the
problem was studied by Trong-Phuong [16] in the
case of m 1 ; however, the problem has more
challenge with m , m 2 .
The rest of our paper consists of three sections.
In Section 2, we establish our estimator. In Section
3, we state main results of our paper. Finally, some
conclusions are presented in Section 4.
For convenience, we introduce some notations.
For two sequences un and vn of positive real
77
replaced
by
r t g ft (t ) / max g ft (t ) ; t
f mft (t )
in
the
quantity
,
called
the
(t ) r t hft (t ) .
m
form
Nevertheless, the function (t ) depends on the
Fourier transform hft (t ) , which is an unknown
quantity, and so, we cannot use (t ) to estimate
f mft (t ) . Fortunately, from the i.i.d. observations
, Yn , we can estimate hft (t ) by the empirical
Y1 ,
un / vn
characteristics
k-
a
ridge function. Here a 1/ m is a given
parameter, and 0 is a regularization parameter
that will be chosen according to n later so that
0 as n . We then obtain an estimator for
numbers, we write un O vn if the sequence
is bounded. The number of
the
2
hˆft (t ) n1
function
n
j 1
e
itY j
.
combinations from a set of p elements is denoted
Hence, another estimator for f mft (t ) is proposed by
by C pk . The number A is the Lebesgue
(t ) r t hˆft (t ) . Finally, using the Fourier
measure of a set A . For a function
p
Lp , 1 p , the symbol
represents the usual Lp
function
:
Z x
: x 0
supp
-norm
,
of . For a
we
define
\ Z , the closure in
L1
ft
2
x 2 x
ft
L2
ft
,
inversion formula, we derive an estimator for f m
in the final form
1
fˆm , x :
2
of the set
and
for
x
,
1
2
Note
2 2 , which is called the Parseval
identity.
L
e itx (t )dt
that
the
m
dt.
a 1/ m
condition
(3)
implies
almost surely. Thus, the
estimator fˆm, x is well-defined for all values of
x , and moreover, fˆm , belongs to L2 .
1
moreover,
g ft t hˆ ft t
itx
e
2
a
ft
max g t ; t
and
\ Z . Regarding the Fourier transform, we
recall that
m
2
L
2 METHODS
3 RESULTS
We now describe the method for constructing an
estimator for f m . First, from the equation (2) we
In this section, we consider consistency and
convergence rate of the estimator fˆm , given in (3)
f mft (t ) [ f ft (t )]m .
Also,
from
the
independence of X and Z , we obtain h f g ,
where h is density of Y . The latter equation gives
hft t f ft t g ft t , so f mft (t ) [hft (t ) / g ft (t )]m
under
if g ft (t ) 0 . Then applying the Fourier inversion
following proposition.
Proposition 1. Let fˆ
have
formula, we can obtain an estimator for f m .
However, it is very dangerous to use
[hft (t ) / g ft (t )]m as an estimator for f mft (t ) in case
g ft can vanish on
. In this case, to avoid
division by numbers very close to zero, 1/ g ft (t ) is
the mean integrated squared error
2
MISE fˆm, , f m
fˆm, f m . First, a general
bound for
MISE fˆm, , f m
m ,
2
is given in the
, m 1 , be as in (3) with
a 1/ m and 0 1 . Suppose that f L2
Then we have
.
78
SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018
2
1
MISE fˆm, , f m
C
2 1
Cmk k
k 1 n k
m
m
2
ft
4m2k
f ft t
max g ft t ; t
where Ck 72k
2k
2
a
m
1 f ft t
2m
dt
I1
2 m k
a
2k / (2k 1)
, k 1,
k
m
ft
f t dt
(5)
2
dt ,
2m
2
m
2
ft
ft
g t f t
2
a
ft
max g t ; t
2m
max g t ; t
g ft t
g ft t
max g t ; t
,m .
2
ft
a
1 f ft t
m
2m
dt ,
Proof. Since f is a density and is in L2
fm L
1
deduce
,
2
L
f
so
ft
m
, we
L .
g ft t
2m
Using the Parseval identity, the Fubini theorem
and the binomial theorem, we obtain
1
MISE fˆm, , f m
2
1
2
fˆmft, t f mft t dt
2
m
m
ft
f t dt
1
2
g ft t
2
a
ft
max g t ; t
C hˆ t h t h t
ft
ft
k
ft
mk
k 0
f ft t dt.
m
Using the inequality z1 z2 2 z1 2 z2
2
z1 , z2
2
2
2
1
MISE fˆm, , f m I1 I 2 ,
with
(4)
where
I1
I2
g ft t
2
a
ft
max g t ; t
g ft t
2
a
ft
max g t ; t
m
ft m
ft
h t f t dt ,
m
C hˆ t h t h t
ft
ft
k
ft
m
a
2m
Cmk
m
C C
k
m
k 1
g ft t
m
4m2k
f ft t
2
U j n1 e
hˆft t hft t 2 k hft t 2 m k dt
k
m
k 1
max g ft t ; t
k 1
itY j
2 m k
a
2m
n
1 itY
e
j 1 n
j
e ,
j
itY j
j 1,
itY j
U
e
2k
dt.
,n .
satisfies the
j 1, , n
conditions of Lemma A.1 in Chesneau et al. [15],
and moreover, U j 2 / n . Hence, applying
mk
2k
n
Uj
j 1
k
2k
2k
36k
2k 1
2 k 2k
36k
2k 1
m
k
k
n
Uj
2
j 1
k
k
1
1
2 k 2k
4
72k
k : Ck k .
n
n
2k 1 n
C
I 2 2 1 C kk
n
k 1
dt.
k 1
Since hft t f ft t g ft t and g ft t g ft t ,
in which g ft t denotes the conjugate of g ft t ,
we have
max g ft t ; t
2
m
2
m
k
m
2m
2m
Thus,
2
m
Lemma A.1 in Chesneau et al. [15] with
p 2k 1 , we get
yields
g ft t
a
2
k
mk
m k ˆft
ft
ft
Cm h t h t h t dt
k 1
Clearly, the sequence
2
m
m
2
Define
k
m
2m
max g ft t ; t
2
g ft t
m
hˆft t hft t hft t f ft t dt
2
a
ft
max
g
t
;
t
2m 1
m
1
2
I2
2
ft
ft
g t hˆ t
2
a
ft
max g t ; t
g ft t
2
k
m
g ft t
4m2k
f ft t
max g t ; t
ft
2
2 m k
a
2m
dt. (6)
From (4) – (6), we obtain the desired conclusion.
Proposition 2. Let the assumptions of Proposition
1 hold. Then there exists a k0 0 depending only
on g such that
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m
C
C kk
n
k 1
4m2k
g ft t
k
m
f ft t
max g ft t ; t
22 m 1 k0
2
1
t k0
t
ma
a
Proof. Since Z g ft 0 and the Lebesgue
2 m k
2m
79
dt
dominated convergence theorem, we get
m
1
dt Cmk Ck m .
k 1
n
2
.
g t
ft
max g ft t ; t
2
g ft 0 1 , there is a constant
k0 0
depending only on g such that g ft t 1/ 2 for
all t k0 . Then for k 1,
g ft t
max g ft t ; t
2
max
g ft t
g t ; t
g t
ft
22 m 1 k0
Hence,
m
2m
a
dt
1
t k0
C
Cmk kk
n
k 1
f t
2m
t
ma
22 m 1 k0
22 m 1 k0
t k0
dt
2m
hft t
2 m k
dt
1
max g ft t ; t
2
a
m
hft t
2 m k
dt
g t
max g t ; t
ft
2
a
m
1 f ft t
2m
dt
f ft t
max g t ; t
ft
2
t k0
In the rest of this section, we study rates of
convergence of MISE fˆm, , f m . To do this, we
2 m k
a
2m
F , L density on
and
a 1/ m and is a positive parameter
depending on n such that 0 and n m
as n . Then MISE fˆm, , f m 0 as n .
2
with 1/ 2 , L 0 . The class F , L contains
Z g ft 0 . Let fˆm , be as in (3), where
ft t 1 t 2 dt L,
2
given in the following theorem.
u
The mean consistency of the estimator fˆm , is
f L2
:
sup ft u 1 u 2 L
Theorem 3. Suppose that
need prior information for f and g . Concerning
the density f , we assume that it belongs to the
class
dt
1 m kC
m Cm kk
dt
ma
k 1
n
t
1 m k 1
dt Cm Ck m .
ma
k 1
t
n
1
t k0
so Z g ft 0 .
4m2k
Theorem 3 is satisfied for normal, gamma,
Cauchy, Laplace, uniform, triangular densities,
among others. In particular, if the noise density g
is a compactly supported, the Fourier transform
g ft can be extended to an analytic function on .
This implies the set Z g ft is at most countable,
The proof of the proposition is completed.
\Z g
ft
g ft t
2m
ft
Remark 4. The condition Z g ft 0 in
1
dt m .
dt
Combining this with Proposition 1, Proposition 2
and the assumptions of the present theorem, we
obtain the conclusion.
2m
2
ft
1
t k0
a
2 m k
2m
0 as n .
, m we have
4m2k
ft
a
1 f ft t
m
2
Proof. Since the function g ft is continuous on
and
2m
many important densities, for example, normal and
Cauchy
densities.
Note
that
F , L L1 L2 . In fact, for positive
integer , if a density is in L2
l
weak derivatives , l 1,
2
derivatives are also in L
having
, , and the weak
, then
belongs to
F , L for L 0 large enough. Regarding the
noise density g , we consider the following classes
of g :
80
SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018
F M density on
: L2
,
Theorem 6. Let 1/ 2 , L 0 . Assume that
g F M with M 0 . Let fˆ be as in (3) for
supp M , M ,
m ,
F c1 , c2 , d ,
: c1e d t ft t c2 e d t , t
density on
a known a 1/ m and n with 0 1/ m .
Then
we
have
m
ˆ
sup f F , L MISE f m, , f m O ln n
.
,
in which M , c1 , c2 , d , are positive constants.
Proof.
The class F M includes compactly supported
densities on M , M . The class F c1 , c2 , d ,
contains densities in which Fourier transforms
converge to zero with exponential rate of order .
Normal and Cauchy densities are typical examples
of F c1 , c2 , d , . In fact, using the Fourier
inversion formula and the Lebesgue dominated
convergence theorem, one can show that each
element of F c1 , c2 , d , is an infinitely
30 2m 1 Me4
2eMR 1 ln R ln 15e ln
for 0 small enough we have
30 1 Me4
In addition,
BR, t
1/ 2
ln 1
.
1/ 2
.
ln n
1/ 2
R 2eM ln n
1
1/ 2
2
J :
g t
ft
g ft t t
t R , g ft t
a
2
f ft t
2m
f ft t
t R , g ft t t
t R
a
m
1 f ft t
t R , g ft t t
2m
dt
dt
2m
f ft t
a
where
t
2m
max g ft t ; t
BR ,
dt
2m
f ft t
f ft t
2m
dt 2 R 2 m
t R
f ft t
note
ft
2m
dt
dt
we
: t R, g
a
t
t
a
2m
dt ,
that
.
Moreover, since f F , L , we derive
t R
R0
that
Then
take
for 0 R a 2 , we have
Lemma 5. Suppose g F M . Given 1 . For
1
We
, and BR, 2 R 2 m for n large enough. Now,
convergence rate established in Chesneau et al.
[15].
3
f F , L .
there exists an R 0 depending on n such that
et al. [15]. The reason for considering this class in
the present paper is that we want to demonstrate
that the estimator fˆm , can also be attained the
0 small enough, we choose an
depending
on
such
Suppose
with
2m 1 , 0 1/ m and n
0 / 2 . Then applying Lemma 5 gives that
smooth” densities. In fact, the case of
g F c1 , c2 , d , has been studied in Chesneau
stating main result of our paper, we need the
following auxiliary lemma. This auxiliary lemma
is not a new result. It is quite similar to Theorem 3
in Trong-Phuong [16].
differentiable
function
on
.
Hence,
F c1 , c2 , d , is often called the class of “super-
Now, we consider the case g F M . Before
| f ft (t ) |2 m dt
t R
| f ft (t ) |2 (1 t 2 ) [| f ft (t ) |2 (1 t 2 ) ]m 1 (1 t 2 ) m dt
Lm R 2 m .
Hence,
J 2 Lm R 2 m
R 2eM ln 1
1
we have BR , 2 R , where
2 Lm 30 2m 1 Me 4
O ln n
m
.
m
ln n
m
(7)
: t R, g
ft
t .
Main result of our paper is the following
theorem.
Combining (7) with Proposition 1 and Proposition
2,
we
obtain
MISE( fˆm, , f m ) O (ln n) m (n m )1 .
Now,
we need to choose 0 according to n so that
R a 2 , and rate of convergence of (n m )1 is
faster than that of (ln n) m . A possible choice is
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n . Then the conclusion of the theorem is
followed.
Remark 7. The parameter in Theorem 6 does
not depend on , the prior degree of smoothness
of f . Therefore, the estimator fˆ x can be
m ,
computed with out any knowledge concerning the
degree of smoothness.
Finally,
we
consider
the
case
g F c1 , c2 , d , . We have
Theorem 8. Let 1/ 2 , L 0 . Suppose that
g F c1 , c2 , d , , where c1 , c2 , d , are the
given positive constants. Let fˆm , be as in (3) for a
a 1/ m
known
n
18 m / 16m2
1 4 am / 4m
ln n
and
. Then we have
2 m /
sup f F , L MISE fˆm, , f m O ln n
.
Proof. Suppose f F , L . Let T be a positive
number that will be selected later. Using the
inequality
0 max g ft t ; t
2
a
m
g ft t
2m
m t
am
m
C
Q :
C kk
n
k 1
Cm
nm
m 1
f F , L , we have
k 1
J :
g
Q2
2m
t
2m t
c1
4 m
t T
ft
2
2 am
t
t T
f ft t
4m
f ft t
2m t
2
a
2 am
1 t
2
a
2m
dt
e 4 md t
f
t
ft
t
t T
2m
m 1
C
k
m
Ck
k
n
c2
dt
dt
ft
m 1
C
k
m
f
t T
a
2m
dt
dt
2m
2 m k
f ft t
t ;
2
a
t
2m
dt : Q1 Q2 .
4m2k
2m
k
m
t T
t T
dt
4m2k
t T
2m
e 2 kd t
ft
f
2 m k
2 am
t
f
e 4 m 2 k d t
t
ft
t
f
ft
2 m k
t
t
dt
dt
2 m k
2 am
dt
Ck
2 k
L c1 e 2 kdT
nk
m 1
C
2 m k
t
2k
ft
g t
ft
g ft t
L c2
2m
Ck
2 k
c1
nk
dt
f ft t
t
4m2k
max g
k 1
Ck L max
k 1
4m2k
1
2m
c
2 k
1
e 4 m 2 k dT T 2 am
; c2
4m2k
1
1
k e 2 kdT k 2 m e 4 m 2 k dT T 2 am .
n
n
f ft t dt
2
f ft t 2 1 t 2
Cmk
k 1
f
2m
O 2 m e 4 mdT T 2 am T 2 m .
Also,
1
m
2 am
max g ft t ; t
g
2
m 1
2m
t ;
2
g ft t
a
2m
k 1
ft
ft
2
2m
g ft t
Cm
1
Q1 m
dt
dt
2 am
t T
n t T g ft t 2 m
2m t
1
T
O m e 2 mdT m 2 m T 1 2 am e 2 mdT ,
n
n
2m
max g ft t ; t
t T
t
2 m k
For the quantities Q1 and Q2 , we have the
estimates
2
ft
Ck
nk
Cmk
f ft t
max g ft t ; t
g ft t
max g
4m2k
g ft t
k
m
for
and the assumptions g F c1 , c2 , d , ,
all t
81
m 1
1 t
2
m
dt
Combining Proposition 1 with the estimates of J ,
Q1 and Q2 , we get
MISE fˆm , , f m O 2 m e 4 mdT T 2 am T 2 m
T 2 mdT
1
e
m 2 m T 1 2 am e 2 mdT
m
n
n
C C L max c
m 1
k
k 1
k
m
1
2 k
; c2
4m2k
1
1
k e 2 kdT k 2 m e 4 m 2 k dT T 2 am .
n
n
82
SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018
Choosing T ln n / 8md
1/
MISE
fˆ
m ,
O n
1/ 2
, fm
ln n
1
n m 1/ 4
2 am /
ln n
1/
m 1
C
k
m
Ck L max
k 1
2 m ln n
c
2 m /
1
n m 1/ 4 2 m
2 k
1
; c2
ln n
1 2 am /
4m2k
1
n m 1/ 4
2 am /
ln n
1/
m 1
C
k
m
Ck L max
k 1
2 m ln n
c
2 k
; c2
O n
ln n
1
n 3/ 2 1/ 4 m
2 am /
ln n
2m
1 2 am /
1 2 am /
4m2k
Choosing n
ln n
2 am /
ln n
1
1
O 11/ 4 m 3/ 2 1/ 4 m 2 m
n
n
1/ 2
ln n
2 m /
1
n m 1/ 4 2 m
1
REFERENCES
ln n
2 am /
1
.
1 4 am / 4 m
ln n
[2].
L.A. Stefanski, R.J. Carroll, “Deconvoluting kernel
density estimators”, Statistics, 21, pp. 169–184, 1990.
[3].
J. Fan, “On the optimal rates of convergence for
nonparametric deconvolution problems”, The Annals of
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[4].
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implies
the desired conclusion.
Remark 9. We see that the convergence rate of
uniformly over the class
MISE fˆm, , f m
R.J. Carroll, P. Hall, “Optimal rates of convergence for
deconvolving a density”, Journal of American Statistical
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2 m /
2m
18 m / 16 m2
[1].
1
1
O k 11/ 4 m 1/ 2 k 11/ 4 m 2 m
n
n
O n1/ 2 ln n
unknown noise density g. We leave this problem
for our future research.
yields
F , L in Theorem 8 is as same as that of
Chesneau et al. [15] when g F c1 , c2 , d , . In
particular, when m 1 , the convergence rate also
coincides with the optimal rate of convergence
proven in Fan [3].
4 CONCLUSIONS
We have considered the problem of
nonparametric estimation of the
m-fold
convolution fm in the additive noise model (1),
where the noise density g is known and assumed to
be compactly supported. An estimator for the
function fm has been proposed and proved to be
consistent with respect to the mean integrated
squared error. Under some regular conditions for
the density f of X, we derive a convergence rate of
the estimator. We also have shown that the
estimator attains the same rate as the one of
Chesneau et al. [15] if the density g is
supersmooth. A possible extension of this work is
to study our estimation procedure in the case of
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TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018
CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018
83
Ước lượng một tự tích chập trong một mô
hình cộng nhiễu với hàm mật độ nhiễu
có giá compact
Cao Xuân Phương
Trường Đại học Tôn Đức Thắng
Tác giả liên hệ:
Ngày nhận bản thảo: 06-05-2017, ngày chấp nhận đăng: 15-05-2017, ngày đăng: 10-08-2018
Tóm tắt – Bài báo này đề cập mô hình Y X Z ,
trong đó Y là một biến ngẫu nhiên quan trắc được,
X là một biến ngẫu nhiên không quan trắc được
với hàm mật độ f chưa biết, và Z là nhiễu ngẫu
nhiên độc lập với X . Hàm mật độ g của Z được
giả thiết biết chính xác và có giá compact. Bài báo
nghiên cứu vấn đề ước lượng phi tham số cho tự
f f f m
tích chập m
( lần) trên cơ sở mẫu
Y,
,Y
n
quan trắc 1
độc lập, cùng phân phối được
lấy từ phân phối của Y . Dựa trên các quan trắc
này cũng như phương pháp chỉnh hóa tham số
f
chóp, một ước lượng cho m phụ thuộc vào hai
tham số chỉnh hóa được đề xuất, trong đó một
tham số được cho trước và tham số còn lại sẽ được
chọn sau. Ước lượng này được chứng tỏ là vững
tương ứng với trung bình sai số tích phân bình
phương dưới một số điều kiện cho các tham số
chỉnh hóa. Sau đó, nghiên cứu tốc độ hội tụ của
ước lượng dưới một số giả thiết chính quy bổ sung
cho hàm mật độ f .
Từ khóa – Ước lượng, hàm mật độ nhiễu có giá compact, tốc độ hội tụ
