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Estimation of a fold convolution in additive noise model with compactly supported noise density

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 (xphuongcao@gmail.com)

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 )  n1

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

4m2k

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

mk

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

4m2k

f ft  t 



2

U j  n1 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

mk






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
mk 
 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 



4m2k

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

4m2k

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 .

4m2k



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

4m2k
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



R0
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


TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018
CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018

  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

18 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 .

4m2k
2m



k
m



t T

t T

dt

4m2k



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

4m2k

max g



k 1

Ck L max

k 1

4m2k

1

 2m

 c 

2 k

1



e   4 m  2 k  dT T 2 am 


;  c2 

4m2k





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

4m2k

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 

4m2k





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  / 

4m2k






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  / 

4m2k

Choosing   n

  ln n 

2 am / 

 ln n 

1
1

 O  11/  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
Statistics, 19, pp. 1257–1272, 1991.

[4].

M.H. Neumann, “On the effect of estimating the error
density in nonparametric deconvolution”, Journal of
Nonparametric Statistics, 7, pp. 307–330, 1997.

[5].

M. Pensky, B. Vidakovic, “Adaptive wavelet estimator
for nonparametric density deconvolution”, The Annals of
Statistics, 27, pp. 2033–2053, 1999.

[6].

P. Hall, A. Meister, “A ridge-parameter approach to
deconvolution”, The Annals of Statistics, 35, pp. 1535–
1558, 2007.

[7].

C. Butucea, A.B. Tsybakov, “Sharp optimality in density
deconvolution with dominating bias”, Theory Probability
and Applications, 51, pp. 24–39, 2008.

[8].

J. Johannes, “Deconvolution with unknown error
distribution”, The Annals of Statistics, 37, pp. 2301–2323
2009.

[9].

E.W. Frees, “Estimating densities of functions of
observations,” Journal of the American Statistical
Association, 89, pp. 517–525, 1994.

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
Association, vol. 83, pp. 1184–1186, 1988.

2 m / 

 2m 

18 m / 16 m2 

[1].



1
1

 O  k 11/  4 m    1/ 2  k 11/  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

[10]. A. Saavedra, R. Cao, “On the estimation of the marginal
density of a moving average process”, The Canadian
Journal of Statistics, 28, pp.799–815, 2000.
[11]. I.A. Ahmad, Y. Fan, “Optimal bandwidth for kernel
density estimator of functions of observations”, Statistics
& Probability Letters, 51, pp. 245–251, 2001.
[12]. I.A. Ahmad, A.R. Mugdadi, “Analysis of kernel density
estimation of functions of random variables”, Journal of
Nonparametric Statistics, vol. 15, pp. 579–605, 2003.
[13]. C. Chesneau, F. Comte, F. Navarro, “Fast nonparametric
estimation for convolutions of densities”, The Canadian
Journal of Statistics, vol. 41, pp. 617–636, 2013.
[14]. C. Chesneau, F. Navarro, “On a plug-in wavelet
estimator for convolutions of densities”, Journal of
Statistical Theory and Practice, vol. 8, pp. 653–673,
2014.
[15]. C. Chesneau, F. Comte, G. Mabon, F. Navarro,
Estimation of convolution in the model with noise,
Journal of Nonparametric Statistics, vol. 27, pp. 286–
315, 2015.
[16]. D.D.
Trong,
C.X.
Phuong,
Ridge-parameter
regularization to deconvolution problem with unknown
error distribution, Vietnam Journal of Mathematics, vol.
43, pp. 239–256, 2015.


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: xphuongcao@gmail.com
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



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