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

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018

CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018

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

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018

CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018

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

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

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

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

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

[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

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

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018

CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018

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

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018

CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018

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

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

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

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

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

[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

## Estimation of Proper Strain Rate in the CRSC Test Using a Artificial Neural Networks

## Sensor-based navigation of a mobile robot in an indoor environment

## Tài liệu Maaref-Sensor-based navigation of a mobile robot in an indoor environment docx

## Tài liệu Báo cáo Y học: Exploring the role of a glycine cluster in cold adaptation of an alkaline phosphatase pdf

## Báo cáo Y học: The presence of a helix breaker in the hydrophobic core of signal sequences of secretory proteins prevents recognition by the signal-recognition particle in Escherichia coli doc

## Báo cáo "The linguitic Situation of a Hmong Community in the North - West of Vietnam " pdf

## The Project Gutenberg EBook of A First Book in Algebra, pot

## Báo cáo khoa học: The antagonistic effect of hydroxyl radical on the development of a hypersensitive response in tobacco pot

## Báo cáo khoa học: Contribution of a central proline in model amphipathic a-helical peptides to self-association, interaction with phospholipids, and antimicrobial mode of action ppt

## Báo cáo khoa học: Predicting the substrate speciﬁcity of a glycosyltransferase implicated in the production of phenolic volatiles in tomato fruit pptx

Tài liệu liên quan