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 .
Trang 1Estimation of a fold convolution in
additive noise model with compactly
supported noise density
Cao Xuan Phuong
Abstract – Consider the model Y X Z,
is an unobservable random variable with
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
well as the ridge-parameter regularization
method, we propose an estimator for the
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
n this paper, we consider the additive noise
model
YXZ (1)
whereYis an observable random variable, Xis an
unobservable random variable with unknown
density f , andZis an unobservable random noise
with known densityg The densitygis called
noise density We also suppose thatXandZ are
independent Estimating f on basis of i.i.d
Received 06-05-2017; Accepted 15-05-2017; Published
10-8-2018
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, ,Y n 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
times
m m
f f f m , (2) based on the observations In the free-error case, i.e Z0, 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 m1, the problem of estimating f m
reduces to the density deconvolution problem To the best of our knowledge, for m ,m2, 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 munder the assumption that ft
g is nonvanishing on , where the function ft itx
g t f x e 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
I
Trang 2instance, 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 densityg In fact, the
problem was studied by Trong-Phuong [16] in the
case of m1; however, the problem has more
challenge with m ,m2
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 u n and v n of positive real
numbers, we write u n O v n if the sequence
u n/v n is bounded The number of k
-combinations from a set of p elements is denoted
by k
p
C The number A is the Lebesgue
measure of a set A For a function
p
L
, 1 p , the symbol p
represents the usual p
L -norm of For a
supp \Z , the closure in of the set
\ Z Regarding the Fourier transform, we
recall that ft ft
2
ft
2
, which is called the Parseval
identity
2 METHODS
We now describe the method for constructing an
estimator for f m First, from the equation (2) we
( ) [ ( )]m
m
f t f t Also, from the
independence of X and Z , we obtain h f g,
where h is density of Y The latter equation gives
ft ft ft
( ) [ ( ) / ( )]m m
f t h t g t
if ft
g t Then applying the Fourier inversion
formula, we can obtain an estimator for f m
However, it is very dangerous to use
[ ( ) / ( )]m
h t g t as an estimator for ft
( )
m
f t in case
ft
g can vanish on In this case, to avoid
division by numbers very close to zero, ft
1/g ( )t is
ridge function Here a1/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 ft
( )
m
f t in the form ft
t r t h t m Nevertheless, the function ( )t depends on the Fourier transform ft
( )
h t , which is an unknown
quantity, and so, we cannot use ( )t to estimate ft
( )
m
f t Fortunately, from the i.i.d observations
1, , n
Y Y , we can estimate ft
( )
h t by the empirical
characteristics function ft 1
1
j
Hence, another estimator for ft
( )
m
f t is proposed by
ˆft
m
t r t h t Finally, using the Fourier inversion formula, we derive an estimator for f m
in the final form
,
2 ft
1
2
ˆ 1
itx m
m itx
a
(3)
Note that the condition a1/m implies
almost surely Thus, the estimator ˆ,
m
f x is well-defined for all values of
x , and moreover, ˆ,
m
f belongs to 2
3 RESULTS
In this section, we consider consistency and convergence rate of the estimator ˆ,
m
f given in (3) under the mean integrated squared error
2
MISE f m,f m f m f m First, a general bound for MISEˆ, ,
m m
f f is given in the following proposition
m
f , m1, be as in (3) with
1/
a m and 0 1 Suppose that 2
Then we have
Trang 3
2 2
ft
2 ft
ft
4 2 2
ft ft
2 2 ft 1
1
ˆ
2 1
,
m
m
a
m
m
k k
k
C
n
72 k 2 / (2 1)k
k
C k k k , k1, ,m
Proof Since f is a density and is in 2
deduce 1 2
m
m
Using the Parseval identity, the Fubini theorem
and the binomial theorem, we obtain
2
ft ft , ,
2
ft ft
ft 2
ft
2 ft
ft ft ft ft 2
ft
ft
ft f 2
ft
1
MISE ,
2
ˆ
1
m
m
a
m
m
a
m
k m a
g t h t
f t dt
g t
g t
C h t h
2
t ft ft 0
.
k
Using the inequality z1z2 22z122z22 with
1, 2
z z yields
1 ˆ
(4) where
2 ft
ft
m
a
2 ft
ft
1
m
m k k
m a k
g t t
Since ft ft ft
h t f t g t and ft ft
g t g t ,
in which ft
g t denotes the conjugate of ft
g t ,
we have
2 2
ft
ft
2 2
ft
2 ft 2
ft
m
m a
m
m m
a
(5)
ft
ft ft ft
2 ft
2 2
ft ft ft 2
2
ft ft
2 2
ˆ
ˆ
1
2 1
m
m k k
m m
m
m k
m
n itY itY
a j
g t
g t
n
1
m k
dt
Define 1 itY j itY j
j
, j1, ,n Clearly, the sequence U j j1, ,n satisfies the conditions of Lemma A.1 in Chesneau et al [15], and moreover, U j 2 /n Hence, applying Lemma A.1 in Chesneau et al [15] with
p k , we get
2
2 2
2 36
2 1
k
k
k
k
Thus,
4 2 2
ft ft
ft 1
k
C
n
g t t
(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
Trang 4
0
2 2 ft 1
2 1
0
1
m
k k
a k
m
m k
t k
k
C
n
n t
Proof Since the function gft is continuous on
and ft
g , there is a constant k0 0
depending only on g such that ft
1/ 2
g t for all t k0 Then for k1, ,m we have
0
4 2
ft
2 ft 2 2
ft
2
ft
2 ft 2 2
ft
2 ft 2
2 1
0
m k
m k m
a
m
m k m
a
m k
m
ma m
t k
g t
g t
t
Hence,
0
0
2 2 ft 1
2 1
0
1
2 1
0
1
2
m k m k m
k k
a k
m
m
t k
k m
m k
t k
k
C
n
C
n t
n t
The proof of the proposition is completed
The mean consistency of the estimator ˆ ,
m
f is given in the following theorem
f L and
ft
0
Z g
Let ˆ,
m
f be as in (3), where
1/
a m and is a positive parameter
depending on n such that 0 and m
n
as n Then MISEˆ , , 0
m m
f f as n
Proof Since ft
0
Z g
and the Lebesgue dominated convergence theorem, we get
ft
2 2
ft
2 ft 2
ft
2 2
ft
2 ft 2
1
1
m
m m
a
m
m m
as n
Combining this with Proposition 1, Proposition 2 and the assumptions of the present theorem, we obtain the conclusion
0
Z g
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 ft
g can be extended to an analytic function on This implies the set ft
Z g is at most countable,
so ft
0
Z g
In the rest of this section, we study rates of convergence of MISEˆ , ,
m m
f f To do this, we need prior information for f and g Concerning the density f , we assume that it belongs to the class
2
2
u
F
with 1/ 2, L0 The class F, L contains many important densities, for example, normal and
integer , if a density is in 2
L having weak derivatives l , l1, ,, and the weak derivatives are also in 2
L , then belongs to
, L
F for L0 large enough Regarding the noise density g, we consider the following classes
of g:
Trang 5
2
M M
F
, , ,
c c d
F
in which M c c d, ,1 2, , are positive constants
The class F M includes compactly supported
densities on M M, The class Fc c d1, 2, ,
contains densities in which Fourier transforms
converge to zero with exponential rate of order
Normal and Cauchy densities are typical examples
of Fc c d1, 2, , In fact, using the Fourier
inversion formula and the Lebesgue dominated
convergence theorem, one can show that each
element of Fc c d1, 2, , is an infinitely
differentiable function on Hence,
c c d1, 2, ,
F is often called the class of
“super-smooth” densities In fact, the case of
1, 2, ,
gF c c d has been studied in Chesneau
et al [15] The reason for considering this class in
the present paper is that we want to demonstrate
that the estimator ˆ,
m
f can also be attained the convergence rate established in Chesneau et al
[15]
Now, we consider the case gF M Before
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]
0
small enough, we choose an R0
2eMR 1 lnRln 15e ln Then
for 0 small enough we have
30 1 Me ln R 2eM ln
In addition, we have B R, 2R, where
R
Main result of our paper is the following
theorem
gF M with M 0 Let ˆ ,
m
f be as in (3) for
a known a1/m and n with 0 1/ m
supfFL MISE f m,f m O lnnm
Proof Suppose f F,L We take
, 0 1/ m and n with
0 / 2 Then applying Lemma 5 gives that there exists an R0 depending on n such that
R
for n large enough Now,
R
, we have
ft
ft
2 2
ft
2 ft 2
ft
2 ft
2 ft ,
,
a
a a
m
m m
a m
g t t
m
t R g t t
R
g t
f t dt
f t dt
Moreover, since fF,L, we derive
ft 2
ft 2 2 ft 2 2 1 2
2
| ( ) |
| ( ) | (1 ) [| ( ) | (1 ) ] (1 )
m
t R
t R
L R
Hence,
2
4 2
m m
m m
Combining (7) with Proposition 1 and Proposition
, ˆ MISE( , ) (ln ) m ( m)
m m
f f O n n Now,
we need to choose 0 according to n so that
2
a
R
, and rate of convergence of 1
( m)
n
is faster than that of (ln )m
n A possible choice is
Trang 6 Then the conclusion of the theorem is
followed
not depend on , the prior degree of smoothness
of f Therefore, the estimator ˆ,
m
f x can be computed with out any knowledge concerning the
degree of smoothness
Finally, we consider the case
1, 2, ,
1, 2, ,
gF c c d , where c c d1, 2, , are the
given positive constants Let ˆ,
m
f be as in (3) for a
1 8 / 16 2 1 4 / 4
ln
,
supfFL MISE f m,f m O lnnm
Proof Suppose fF,L Let T be a positive
number that will be selected later Using the
inequality
all t and the assumptions gFc c d1, 2, ,,
,
fF L , we have
2 2
ft
2 ft 2
ft
2
2
2 ft 2 2
ft
2
2
ft ft 4
ft
2 2
4 2 4 ft
1
1
ft 2 ft 2 2
2 4 2 2
m
m m
a
am m
m m a
am
m
m
am
t T
t T
t
t
Also,
2 2 ft 1
2 ft
2 2 ft
1
2 2 ft 1
:
m
k k
a k
m m
m m
a
m
k k
a k
C
g t C
dt n
C
For the quantities Q1 and Q2, we have the estimates
2 ft
2
1
1
,
m m
C
T
2 ft 1
ft 1
4 2 2
ft ft
2 2 1
2
2 2 ft 1
1
2
4 2 ft
4 2 2
2 2
1
2 2 1 1
4 2 2
1
m k m
k
am m
m
m k
t T k
m k
am
m
k
k
C
dt t
C
n
c
t C
n
L c
4 2 2 2
1
2 4 2
1 2 1
4 2
2
.
m
m
k
k
Combining Proposition 1 with the estimates of J, 1
Q and Q2, we get
,
2 1
1
4 2
2
ˆ
1
m mdT am m
m m
m
k
k m k
m k dT
T
Trang 7Choosing 1/
T n md yields
,
1
1
2 /
1 1/ 4 1/ 2 1 1/ 4 2
1/
1/ 4
ˆ
ln
1
ln
m m
am
m
k
m k
k
am
m
n
n
1 2 / 1/ 4 2
1
1
2 /
1 1/ 4 3/ 2 1/ 4 2
1 2 / 2 3/ 2 1/ 4
1 ln
ln
am
m
k
m k
k
am
am m m
n n
n
n
Choosing 1 8 / 16 2 1 4 / 4
ln
the desired conclusion
Remark 9 We see that the convergence rate of
MISE f m,f m uniformly over the class
, L
F in Theorem 8 is as same as that of
Chesneau et al [15] when gFc c d1, 2, , In
particular, when m1, 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 f m in the additive noise model (1),
where the noise density g is known and assumed to
be compactly supported An estimator for the
function f m 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
unknown noise density g We leave this problem
for our future research
REFERENCES [1] 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] 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
[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
Trang 8Ước 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
tích chập f m f f ( m lần) trên cơ sở mẫu
quan trắc Y1, ,Y n
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 chóp, một ước lượng cho f 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