Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 604150, 21 pages
doi:10.1155/2011/604150
Research Article
Adaptive Wavelet Estimation of a Biased Density
for Strongly Mixing Sequences
Christophe Chesneau
Université de Caen-Basse Normandie, Département de Mathématiques, UFR de Sciences,
14032 Caen, France
Correspondence should be addressed to Christophe Chesneau, christophe.chesneau@gmail.com
Received 7 December 2010; Accepted 24 February 2011
Academic Editor: Palle E. Jorgensen
Copyright q 2011 Christophe Chesneau. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The estimation of a biased density for exponentially strongly mixing sequences is investigated. We
construct a new adaptive wavelet estimator based on a hard thresholding rule. We determine a
sharp upper bound of the associated mean integrated square error for a wide class of functions.
1. Introduction
In the standard density estimation problem, we observe n random variables X1 , . . . , Xn with
common density function f. The goal is to estimate f from X1 , . . . , Xn . However, in some
applications, X1 , . . . , Xn are not accessible; we only have n random variables Z1 , . . . , Zn with
the common density
gx μ−1 wxfx,
1.1
where
w denotes a known positive function and μ is the unknown normalization parameter:
μ wyfydy. Our goal is to estimate the “biased density” f from Z1 , . . . , Zn . Practical
examples can be found in, for example, 1–3 and the survey by the author of 4.
The standard i.i.d. case has been investigated in several papers. See, for example, 5–
9. To the best of our knowledge, the dependent case has only been examined in 10 for
associated positively or negatively Z1 , . . . , Zn . In this paper, we study another dependent
and realistic structure which has not been addressed earlier: we suppose that Z1 , . . . , Zn
is a sample of a strictly stationary and exponentially strongly mixing process Zi i∈ to
be defined in Section 2. Such a dependence condition arises for a wide class of GARCHtype time series models classically encountered in finance. See, for example, 11, 12 for an
overview.
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We focus our attention on the wavelet methods because they provide a coherent set of
procedures that are spatially adaptive and near optimal over a wide range of function spaces.
See, for example, 13, 14 for a detailed coverage of wavelet theory in statistics. We develop
two new wavelet estimators: a linear nonadaptive based on projections and a nonlinear
adaptive using the hard thresholding rule introduced by 15. We measure their performances
by determining upper bounds of the mean integrated squared error MISE over Besov balls
to be defined in Section 3. We prove that our adaptive estimator attains a sharp rate of
convergence, close to the one attained by the linear wavelet estimator constructed in a
nonadaptive fashion to minimize the MISE.
The rest of the paper is organized as follows. Section 2 is devoted to the assumptions
on the model. In Section 3, we present wavelets and Besov balls. The considered wavelet
estimators are defined in Section 4. Section 5 is devoted to the results. The proofs are
postponed in Section 6.
2. Assumptions on the Model
We assume that Z1 , . . . , Zn coming from a strictly stationary process Zi i∈ . For any m ∈
we define the mth strongly mixing coefficient of Zi i∈ by
am sup
Z
Z
A,B∈F−∞,0
×Fm,∞
|A ∩ B − AB|,
,
2.1
Z
is the σ-algebra generated by the random variables . . . , Zu−1 , Zu
where, for any u ∈ , F−∞,u
Z
and Fu,∞ is the σ-algebra generated by the random variables Zu , Zu1 , . . ..
We consider the exponentially strongly mixing case, that is, there exist three known
constants, γ > 0, c > 0, and θ > 0, such that, for any m ∈ ,
am ≤ γ exp −c|m|θ .
2.2
This assumption is satisfied by a large class of GARCH processes. See, for example, 11, 12,
16, 17.
Note that, when θ → ∞, we are in the standard i.i.d. case.
W.o.l.g., the support of the functions f, and w are 0, 1.
There exist two constants, c > 0 and C > 0, such that
c ≤ inf wx,
x∈0,1
sup wx ≤ C.
x∈0,1
2.3
There exists a known constant C > 0 such that
sup fx ≤ C.
x∈0,1
2.4
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3
For any m ∈ {1, . . . , n}, let gZ0 ,Zm be the density of Z0 , Zm . There exists a constant C > 0
such that
sup
sup
gZ ,Z x, y − gxg y ≤ C.
0 m
m∈{1,...,n} x,y∈0,12
2.5
The two first boundedness assumptions are standard in the estimation of biased
densities. See, for example, 6–8.
3. Wavelets and Besov Balls
Let N be an integer φ and ψ be the initial wavelets of dbN so suppφ suppψ 1 −
N, N. Set
φj,k x 2j/2 φ 2j x − k ,
ψj,k x 2j/2 ψ 2j x − k .
3.1
With an appropriate treatments at the boundaries, there exists an integer τ satisfying 2τ ≥ 2N
such that the collection B {φτ,k ·, k ∈ {0, . . . , 2τ − 1}; ψj,k ·; j ∈ − {0, . . . , τ − 1}, k ∈
{0, . . . , 2j − 1}}, is an orthonormal basis of 2 0, 1 the space of square-integrable functions
on 0, 1. See 18.
For any integer ≥ τ, any h ∈ 2 0, 1 can be expanded on B as
hx 2
−1
j
∞ 2
−1
k0
j k0
α,k φ,k x x ∈ 0, 1,
3.2
hxψj,k xdx.
3.3
βj,k ψj,k x,
where αj,k and βj,k are the wavelet coefficients of h defined by
αj,k 1
hxφj,k xdx,
0
βj,k 1
0
s
M if and only if there
Let M > 0, s > 0, p ≥ 1, and r ≥ 1. A function h belongs to Bp,r
∗
exists a constant M > 0 depending on M such that the associated wavelet coefficients 3.3
satisfy
2τ −1
1/p ⎛ ∞ ⎛
2j −1
1/p ⎞r ⎞1/r
p
p
βj,k ⎠ ⎠ ≤ M∗ .
2τ1/2−1/p
⎝ ⎝2js1/2−1/p
|ατ,k |
k0
jτ
3.4
k0
In this expression, s is a smoothness parameter and p and r are norm parameters. For a
s
M contains some classical sets of functions as the Hölder
particular choice of s, p, and r, Bp,r
and Sobolev balls. See 19.
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4. Estimators
Firstly, we consider the following estimator for μ:
μ
n
1
1
n i1 wZi −1
.
4.1
It is obtained by the method of moments see Proposition 6.2 below.
Then, for any integer j ≥ τ and any k ∈ {0, . . . , 2j − 1}, we estimate the unknown
wavelet coefficient
1
i αj,k 0 fxφj,k xdx by
n φ Z μ j,k
i
,
n i1 wZi 4.2
n ψ Z μ j,k
i
βj,k .
n i1 wZi 4.3
αj,k ii βj,k 1
0
fxψj,k xdx by
Note that they are those considered in the i.i.d. case see, e.g., 8, 9. Their statistical
properties, with our dependent structure, are investigated in Propositions 6.2, 6.3, and 6.4
below.
s
Assuming that f ∈ Bp,r
M with p ≥ 2, we define the linear estimator fL by
fL x j0
2
−1
αj0 ,k φj0 ,k x,
x ∈ 0, 1,
4.4
k0
where α
j,k is defined by 4.2 and j0 is the integer satisfying
1 1/2s1
< 2j0 ≤ n1/2s1 .
n
2
4.5
For a survey on wavelet linear estimators for various density models, we refer the
reader to 20. For the consideration of strongly mixing sequences, see, for example, 21, 22.
We define the hard thresholding estimator fH by
fH x τ
2
−1
j
j1 2
−1
k0
jτ k0
ατ,k φτ,k x βj,k {|βj,k| ≥κλn } ψj,k x,
4.6
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5
x ∈ 0, 1, where ατ,k is defined by 4.2 and βj,k by 4.3, for any random event A, A is the
indicator function on A, j1 is the integer satisfying
n
1
n
< 2 j1 ≤
,
2 ln n11/θ
ln n11/θ
4.7
θ is the one in 2.2, κ is a large enough constant the one in Proposition 6.4 below and λn is
the threshold
λn ln n11/θ
.
n
4.8
The feature of the hard thresholding estimator is to only estimate the “large” unknown
wavelet coefficients of f which contain his main characteristics.
For the construction of hard thresholding wavelet estimators in the standard density
model, see, for example, 15, 23.
5. Results
Theorem 5.1 upper bound for fL . Consider 1.1 under the assumptions of Section 2. Suppose
s
that f ∈ Bp,r
M with s > 0, p ≥ 2, and r ≥ 1. Let fL be 4.4. Then there exists a constant C > 0
such that
1
2
fL x − fx dx
≤ Cn−2s/2s1 .
5.1
0
The proof of Theorem 5.1 uses a suitable decomposition of the MISE and a moment
inequality on 4.2 see Proposition 6.3 below.
Note that n−2s/2s1 is the optimal rate of convergence in the minimax sense for the
standard density model in the independent case see, e.g., 14, 23.
Theorem 5.2 upper bound for fH . Consider 1.1 under the assumptions of Section 2. Let fH be
s
4.6. Suppose that f ∈ Bp,r
M with r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ 1, 2 and s > 1/p}. Then
there exists a constant C > 0 such that
1
0
2
fH x − fx dx
≤C
ln n11/θ
n
2s/2s1
.
5.2
The proof of Theorem 5.2 uses a suitable decomposition of the MISE, some moment
inequalities on 4.2 and 4.3 see Proposition 6.3 below, and a concentration inequality on
4.3 see Proposition 6.4 below.
Theorem 5.2 shows that, besides being adaptive, fH attains a rate of convergence close
to the one of fL . The only difference is the logarithmic term ln n11/θ2s/2s1 .
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Note that, if we restrict our study to the independent case, that is, θ → ∞, the rate of
convergence attained by fH becomes the standard one: log n/n2s/2s1 . See, for example,
14, 15, 23.
6. Proofs
In this section, we consider 1.1 under the assumptions of Section 2. Moreover, C denotes
any constant that does not depend on j, k and n. Its value may change from one term to
another and may depends on φ or ψ.
6.1. Auxiliary Results
Lemma 6.1. For any integer j ≥ τ and any k ∈ {0, . . . , 2j − 1}, let α
j,k be 4.2 and αj,k 1
0 fxφj,k xdx. Then, under the assumptions of Section 2, there exists a constant C > 0 such that
n φ Z 1 1 j,k
i
αj,k − αj,k ≤ C μ
− αj,k − .
n i1 wZi μ
μ
6.1
j,k and βj,k This inequality holds for ψ instead of φ (and, a fortiori, βj,k defined by 4.3 instead of α
1
fxψ
xdx
instead
of
α
).
j,k
j,k
0
Proof of Lemma 6.1. We have
αj,k − αj,k
μ
μ
n φ Z μ
j,k
i
− αj,k
n i1 wZi 1 1
−
.
αj,k μ
μ μ
6.2
Due to 2.3, we have |
μ| ≤ C and |
μ/μ| ≤ C. Therefore
n
φj,k Zi 1 1 αj,k − αj,k ≤ C μ
− αj,k αj,k − .
n i1 wZi μ
μ
6.3
Using 2.4 and the Cauchy-Schwarz inequality, we obtain
αj,k ≤
1
fxφj,k xdx ≤ C
1
0
≤C
φj,k xdx
0
1
2
φj,k x dx
1/2
6.4
C.
0
Hence
n
φj,k Zi 1 1 αj,k − αj,k ≤ C μ
− αj,k − .
n i1 wZi μ
μ
Lemma 6.1 is proved.
6.5
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7
Proposition 6.2. For any integer j ≥ τ such that 2j ≤ n and any k ∈ {0, . . . , 2j − 1}, let αj,k 1
be 4.1. Then,
0 fxφj,k xdx and μ
1 one has
n φ Z μ
j,k
i
n i1 wZi μ1 μ1 ,
αj,k ,
6.6
2 there exists a constant C > 0 such that
n φ Z μ
j,k
i
n i1 wZi 1
≤C ,
n
6.7
3 there exists a constant C > 0 such that
1
1
≤C .
μ
n
These results hold for ψ instead of φ (and, a fortiori, βj,k 6.8
1
0
fxψj,k xdx instead of αj,k ).
Proof of Proposition 6.2. 1 We have
n φ Z μ
j,k
i
n i1 wZi μ
μ
1
0
φj,k Z1 wZ1 μ
1
0
φj,k x
gxdx
wx
φj,k x −1
μ wxfxdx wx
1
6.9
fxφj,k xdx αj,k .
0
Since f is a density, we obtain
1
n
1
1
1
1
1
gxdx
μ n wZ wZ
wx
i
1
0
i1
1
0
1
1
μ−1 wxfxdx wx
μ
1
0
1
fxdx .
μ
6.10
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2 We have
n φ Z μ
j,k
i
n i1 wZi n
n φj,k Zv φj,k Z μ2 wZ , wZ 2
n v1 1
v
v−1
n φj,k Z1 φj,k Zv φj,k Z μ2 μ2
wZ 2 n2
wZ , wZ n
1
v
v2 1
n v−1
φj,k Z1 φj,k Zv φj,k Z μ2
μ2 ≤
wZ 2 n2 wZ , wZ .
n
1
v
v2 1
6.11
Using 2.3 and 2.4, we have supx∈0,1 gx ≤ C. Hence,
φj,k Z1 wZ1 ⎛
≤ ⎝
C
1
φj,k Z1 wZ1 2 ⎞
⎠ ≤ C φj,k Z1 2
2
φj,k x gxdx ≤ C
0
1
6.12
2
φj,k x dx C.
0
It follows from the stationarity of Zi i∈ and 2j ≤ n that
n v−1
n
φj,k Zv φj,k Z φj,k Z0 φj,k Zm wZ , wZ n − m wZ , wZ v2 1
v
0
m
m1
n φj,k Z0 φj,k Zm ≤ n ,
T1 T2 ,
wZ0 wZm m1
6.13
where
φj,k Z0 φj,k Zm T1 n ,
,
wZ0 wZm m1
n φj,k Z0 φj,k Zm ,
T2 n
.
wZ0 wZm j
j
2
−1
m2
Let us now bound T1 and T2 .
6.14
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9
Upper Bound for T1
Using 2.5, 2.3, and doing the change a variables y 2j x − k, we obtain
φj,k Z0 φj,k Zm 1 φj,k x φj,k y
,
gZ0 ,Zm x, y − gxg y
dx dy
wZ0 wZm 0
wx w y
1
φ x φj,k y gZ ,Z x, y − gxg y j,k
≤
dx dy
0 m
wx w y 0
≤C
1
φj,k xdx
2
C 2
−j/2
1
0
φxdx
2
C2−j .
0
6.15
Therefore,
T1 ≤ Cn2−j 2j Cn.
6.16
Upper Bound for T2
By the Davydov inequality for strongly mixing processes see 24, for any q ∈ 0, 1, it holds
that
⎛ ⎛
2/1−q ⎞⎞1−q
φj,k Z0 φj,k Zm q ⎝ ⎝ φj,k Z0 ⎠⎠
,
≤ 10am wZ0 wZ0 wZm ⎛ ⎛ 2 ⎞⎞1−q
φj,k x 2q
φj,k Z0 ⎝ ⎝
⎠⎠ .
sup wZ0 x∈0,1 wx q
≤ 10am
6.17
By 2.3, we have
φj,k x sup ≤ C sup φj,k x ≤ C2j/2
x∈0,1 wx
x∈0,1
6.18
and, by 6.12,
⎛
⎝
φj,k Z0 wZ0 2 ⎞
⎠ ≤ C.
6.19
Therefore,
φj,k Z0 φj,k Zm q
,
≤ C2qj am .
wZ0 wZm 6.20
10
Since
International Journal of Mathematics and Mathematical Sciences
n
m2j
q
mq am ≤
∞
m1
q
mq am γ q
∞
m1
n
T2 ≤ Cn2qj
mq exp−cqmθ < ∞, we have
n
q
am ≤ Cn
q
mq am ≤ Cn.
6.21
n v−1
φj,k Zv φj,k Z wZ , wZ ≤ Cn.
v2 1
v
6.22
m2j
m2j
It follows from 6.13, 6.16, and 6.21 that
Combining 6.11, 6.12, and 6.22, we obtain
n φ Z μ
j,k
i
n i1 wZi 1
≤C .
n
6.23
3 Proceeding in a similar fashion to 2-, we obtain
n
1
1
1
μ n wZ i
i1
v−1 n 1
1
1
1
1
2 2
,
n
wZ1 wZv wZ n v2 1
≤
n 1
1
1
1
1
.
2
,
n
wZ1 n m1
wZ0 wZm 6.24
Using 2.3 which implies supx∈0,1 1/wx ≤ C and applying the Davydov inequality, we
obtain
n
1
1
q
6.25
μ ≤ C n 1 am ≤ C n1 .
m1
The proof of Proposition 6.2 is complete.
Proposition 6.3. For any integer j ≥ τ such that 2j ≤ n and any k ∈ {0, . . . , 2j − 1}, let αj,k 1
fxφj,k xdx and α
j,k be 4.2. Then,
0
1 there exists a constant C > 0 such that
2 1
αj,k − αj,k
≤C ;
n
6.26
2 there exists a constant C > 0 such that
α
j,k − αj,k
4 1
≤ C2j .
n
6.27
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These inequalities hold for βj,k defined by 4.3 instead of αj,k , and βj,k αj,k .
11
1
0
fxψj,k xdx instead of
Proof of Proposition 6.3. 1 Applying Lemma 6.1 and Proposition 6.2, we have
⎛ ⎛
⎞
2 ⎞
2 n φ Z 2 μ
1
1
j,k
i
αj,k − αj,k ≤ C⎝ ⎝ n wZ − αj,k ⎠ μ − μ ⎠
i
i1
C
n φ Z μ
j,k
i
n i1 wZi 1
1
≤C .
μ
n
6.28
2 We have
αj,k − αj,k ≤ αj,k αj,k .
6.29
By 2.3, we have |
μ| ≤ C and supx∈0,1 1/wx ≤ C. So,
n
n φ Z μ
φj,k x 1
φj,k Zi j,k i ≤C
≤ C sup n i1 wZi n i1 wZi wx
x∈0,1
≤ C sup φj,k x ≤ C2j/2 .
6.30
x∈0,1
By 6.4, we have |αj,k | ≤ C. Therefore
αj,k − αj,k ≤ C 2j/2 1 ≤ C2j/2 .
6.31
It follows from 6.31 and 6.28 that
4 2 1
αj,k − αj,k
≤ C2j αj,k − αj,k
≤ C2j .
n
6.32
The proof of Proposition 6.3 is complete.
1
Proposition 6.4. For any j ∈ {τ, . . . , j1 } and any k ∈ {0, . . . , 2j − 1}, let βj,k 0 fxψj,k xdx,
βj,k be 4.3 and λn be 4.8. Then there exist two constants, κ > 0 and C > 0, such that
κλ 1
n
≤ C 4.
βj,k − βj,k ≥
2
n
6.33
Proof of Proposition 6.4. It follows from Lemma 6.1 that
κλ βj,k − βj,k ≥ 2 n ≤ P1 P2 ,
6.34
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where
n
μ ψj,k Zi P1 − βj,k ≥ κCλn ,
n i1 wZi 1 1
P2 − ≥ κCλn .
μ
μ
6.35
In order to bound P1 and P2 , let us present a Bernstein inequality for exponentially strongly
mixing process. We refer to 25, 26.
Lemma 6.5 see 25, 26. Let γ > 0, c > 0, θ > 1 and Zi i∈ be a stationary process such that,
for any m ∈ , the associated mth strongly mixing coefficient 2.2 satisfies am ≤ γ exp−c|m|θ . Let
n ∈ ∗ , h : → be a measurable function and, for any i ∈ , Ui hZi . One assumes that
U1 0 and there exists a constant M > 0 satisfying |U1 | ≤ M < ∞. Then, for any m ∈ {1, . . . , n}
and any λ > 4mM/n, one has
n
1
n
λ2 n
n Ui ≥ λ ≤ 4 exp − 2 4γ exp −cmθ .
m
m 64 U1 8λM/3
i1
6.36
Upper Bound for P1
For any i ∈ {1, . . . , n}, set
Ui μ
ψj,k Zi − βj,k .
wZi 6.37
Then U1 , . . . , Un are identically distributed, depend on the stationary strongly mixing process
Zi i∈ which satisfies 2.2, Proposition 6.2 gives
U1 0,
U12
⎛
ψj,k Z1 ≤ ⎝ μ
wZ1 2 ⎞
⎠≤C
6.38
and, by 2.3 and 6.4,
ψj,k x |U1 | ≤ μ sup βj,k ≤ C sup ψj,k x 1
x∈0,1 wx x∈0,1
≤ C 2j/2 1 ≤ C2j/2 .
6.39
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13
It follows from Lemma 6.5 applied with U1 , . . . , Un , λ κCλn , λn ln n11/θ /n1/2 ,
m u ln n1/θ with u > 0 chosen later, M C2j/2 and 2j ≤ 2j1 ≤ n/ln n11/θ , that
n
1 U ≥ κCλn
P1 n i1 i κ2 λ2n n
n
4γ exp −cmθ
≤ 4 exp −C
m1 κλn M
m
⎛
⎜
≤ 4 exp⎜
⎝−C
4γ
⎞
⎟
κ2 ln n11/θ
⎟
1/2 ⎠
1/θ
11/θ
j/2
/n
1 κ2
u ln n
ln n
n
u ln n1/θ
6.40
exp−cu ln n
2
1/θ
≤ C n−Cκ /u 1κ n1−cu .
Therefore, for large enough κ and u, we have
P1 ≤ C
1
.
n4
6.41
Upper Bound for P2
For any i ∈ {1, . . . , n}, set
Ui 1
1
− .
wZi μ
6.42
Then U1 , . . . , Un are identically distributed, depend on the stationary strongly mixing process
Zi i∈ which satisfies 2.2, Proposition 6.2 gives
U1 0,
U12
≤
1
wZ1 2
≤ C.
6.43
By 2.3, we have
1
1
≤ C.
|U1 | ≤ sup
μ
x∈0,1 wx
6.44
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It follows from Lemma 6.5 applied with U1 , . . . , Un , λ κCλn , λn ln n11/θ /n1/2 ,
m u ln n1/θ with u > 0 chosen later and M C that
n
1
P2 U ≥ κCλn
n i1 i κ2 λ2n n
n
≤ 4 exp −C
4γ exp −cmθ
m1 κλn M
m
⎛
⎞
⎜
≤ 4 exp⎜
⎝−C
4γ
⎟
κ2 ln n11/θ
⎟
1/2 ⎠
1/θ
11/θ
1 κ ln n
/n
u ln n
n
u ln n1/θ
6.45
exp−cu ln n
2
1/θ
≤ C n−Cκ /u n1−cu .
Therefore, for large enough κ and u, we have
P2 ≤ C
1
.
n4
6.46
Putting 6.34, 6.41, and 6.46 together, this ends the proof of Proposition 6.4.
6.2. Proofs of the Main Results
Proof of Theorem 5.1. We expand the function f on B as
fx j0
2
−1
αj0 ,k φj0 ,k x j
∞ 2
−1
βj,k ψj,k x,
x ∈ 0, 1,
6.47
jj0 k0
k0
1
1
where αj0 ,k 0 fxφj0 ,k xdx and βj,k 0 fxψj,k xdx.
We have, for any x ∈ 0, 1,
fL x − fx j0
2
−1
∞ 2
−1
βj,k ψj,k x.
αj0 ,k − αj0 ,k φj0 ,k x −
j
6.48
jj0 k0
k0
Since B is an orthonormal basis of 2 0, 1, we have,
1
0
2
f x − fx dx
L
j0
2
−1
k0
α
j0 ,k − αj0 ,k
2 j
∞ 2
−1
2
βj,k
.
jj0 k0
6.49
International Journal of Mathematics and Mathematical Sciences
15
Using Proposition 6.3, we obtain
j0
2
−1
k0
2 1
αj0 ,k − αj0 ,k
≤ C2j0 ≤ Cn−2s/2s1 .
n
6.50
s
s
M ⊆ B2,∞
M. Hence
Since p ≥ 2, we have Bp,r
j
∞ 2
−1
2
βj,k
≤ C2−2j0 s ≤ Cn−2s/2s1 .
6.51
jj0 k0
Therefore,
1
2
f x − fx dx
L
≤ Cn−2s/2s1 .
6.52
0
The proof of Theorem 5.1 is complete.
Proof of Theorem 5.2. We expand the function f on B as
fx τ
2
−1
ατ,k φτ,k x j
∞ 2
−1
βj,k ψj,k x,
x ∈ 0, 1,
6.53
jτ k0
k0
1
1
where ατ,k 0 fxφτ,k xdx and βj,k 0 fxψj,k xdx.
We have, for any x ∈ 0, 1,
fH x − fx τ
2
−1
j
j1 2
−1
k0
jτ k0
ατ,k − ατ,k φτ,k x βj,k {|βj,k|≥κλn } − βj,k ψj,k x
6.54
j
∞ 2
−1
βj,k ψj,k x.
−
jj1 1 k0
Since B is an orthonormal basis of 2 0, 1, we have
1
0
2
fH x − fx dx
R S T,
6.55
16
International Journal of Mathematics and Mathematical Sciences
where
R
τ
2
−1
ατ,k − ατ,k 2 ,
S
j
j1 2
−1 jτ k0
k0
βj,k {|βj,k|≥κλn } − βj,k
2 ,
6.56
j
∞ 2
−1
2
T
βj,k
.
jj1 1 k0
Let us bound R, T, and S, in turn.
Upper Bound for R
Using Proposition 6.3 and 2s/2s 1 < 1, we obtain
1
≤C
R ≤ C2
n
τ
ln n11/θ
n
2s/2s1
6.57
.
Upper Bound for T
s
s
For r ≥ 1 and p ≥ 2, we have Bp,r
M ⊆ B2,∞
M. Since 2s/2s 1 < 2s, we have
T≤C
∞
2
−2js
≤ C2
−2j1 s
≤C
jj1 1
ln n11/θ
n
2s
≤C
s1/2−1/p
s
For r ≥ 1 and p ∈ 1, 2, we have Bp,r
M ⊆ B2,∞
1/p > s/2s 1. So
T ≤C
∞
ln n11/θ
n
2s/2s1
.
M. Since s > 1/p, we have s 1/2 −
2−2js1/2−1/p ≤ C2−2j1 s1/2−1/p
jj1 1
≤C
6.58
11/θ
ln n
n
2s1/2−1/p
≤C
11/θ
ln n
n
6.59
2s/2s1
.
Hence, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ 1, 2 and s > 1/p}, we have
T ≤C
ln n11/θ
n
2s/2s1
.
6.60
Upper Bound for S
Note that we can write the term S as
S S1 S2 S3 S4 ,
6.61
International Journal of Mathematics and Mathematical Sciences
17
where
S1 j
j1 2
−1 2
βj,k − βj,k {|βj,k|≥κλn} {|βj,k|<κλn/2} ,
jτ k0
S2 j
j1 2
−1 2
βj,k − βj,k {|βj,k|≥κλn} {|βj,k|≥κλn/2} ,
jτ k0
S3 j
j1 2
−1 jτ k0
S4 j
j1 2
−1 jτ k0
6.62
2
βj,k
{|βj,k|<κλn} {|βj,k|≥2κλn} ,
2
{|βj,k|<κλn} {|βj,k|<2κλn} .
βj,k
Let us investigate the bounds of S1 , S2 , S3 , and S4 in turn.
Upper Bounds for S1 and S3
We have
κλ n
,
βj,k < κλn , βj,k ≥ 2κλn ⊆ βj,k − βj,k >
2
κλ κλn
n
⊆ βj,k − βj,k >
,
βj,k ≥ κλn , βj,k <
2
2
βj,k < κλn , βj,k ≥ 2κλn ⊆ βj,k ≤ 2βj,k − βj,k .
6.63
So,
maxS1 , S3 ≤ C
j
j1 2
−1 βj,k − βj,k
jτ k0
2
.
j,k−βj,k |>κλn /2}
{|β
6.64
It follows from the Cauchy-Schwarz inequality, Propositions 6.3 and 6.4, and 2j ≤ 2j1 ≤ n that
βj,k − βj,k
2
{|β
j,k−βj,k |>κλn /2}
κλ 1/2
4 1/2 ≤ βj,k − βj,k > 2 n
1 1/2 1 1/2
1
≤ C 2j
≤ C 2.
n
n
n4
6.65
βj,k − βj,k
18
International Journal of Mathematics and Mathematical Sciences
Since 2s/2s 1 < 1, we have
j1
1
1
1
maxS1 , S3 ≤ C 2 2j ≤ C 2 2j1 ≤ C ≤ C
n
n jτ
n
ln n11/θ
n
2s/2s1
6.66
.
Upper Bound for S2
Using again Proposition 6.3, we obtain
βj,k − βj,k
2 ≤C
1
ln n11/θ
≤C
.
n
n
6.67
Hence,
1 2
−1
ln n11/θ S2 ≤ C
{|βj,k|>κλn/2} .
n
jτ k0
j
j
6.68
Let j2 be the integer defined by
1
2
1/2s1
n
ln n11/θ
<2 ≤
1/2s1
n
j2
ln n11/θ
6.69
.
We have
S2 ≤ S2,1 S2,2 ,
6.70
where
2 2
−1
ln n11/θ C
{|βj,k|>κλn/2} ,
n
jτ k0
j
S2,1
S2,2
j
6.71
j
j1 2
−1
ln n11/θ C
{|βj,k|>κλn/2} .
n
jj2 1 k0
We have
2
ln n11/θ ln n11/θ j2
2 ≤C
≤C
2j ≤ C
n
n
jτ
j
S2,1
ln n11/θ
n
2s/2s1
.
6.72
International Journal of Mathematics and Mathematical Sciences
19
s
s
M ⊆ B2,∞
M,
For r ≥ 1 and p ≥ 2, since Bp,r
S2,2 ≤ C
j
j
j1 2
−1
∞ 2
−1
ln n11/θ 2
2
β
≤
C
βj,k
≤ C2−2j2 s
j,k
nλ2n
jj2 1 k0
jj2 1 k0
≤C
11/θ
ln n
n
6.73
2s/2s1
.
s
For r ≥ 1, p ∈ 1, 2 and s > 1/p, using {|βj,k|>κλn /2} ≤ C|βj,k |p /λn , Bp,r
M ⊆ B2,∞
2s 12 − p/2 s 1/2 − 1/pp 2s, we have
p
S2,2
j
j1 2
−1
ln n11/θ βj,k p ≤ C
≤C
p
nλn
jj2 1 k0
≤C
ln n11/θ
n
2−p/2
s1/2−1/p
∞
M and
2−js1/2−1/pp
jj2 1
6.74
2−p/2
2s/2s1
ln n11/θ
ln n11/θ
−j2 s1/2−1/pp
2
≤C
.
n
n
So, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ 1, 2 and s > 1/p}, we have
S2 ≤ C
ln n11/θ
n
2s/2s1
6.75
.
Upper Bound for S4
We have
j
j1 2
−1
2
βj,k
{|βj,k|<2κλn} .
S4 ≤
6.76
jτ k0
Let j2 be the integer 6.69. Then
S4 ≤ S4,1 S4,2 ,
6.77
where
S4,1 j
j2 2
−1
2
βj,k
{|βj,k|<2κλn} ,
jτ k0
S4,2 j
j1 2
−1
jj2 1 k0
2
βj,k
{|βj,k|<2κλn} .
6.78
We have
S4,1 ≤ C
j2
jτ
2
ln n11/θ ln n11/θ j2
2 ≤C
C
2j ≤ C
n
n
jτ
j
2j λ2n
ln n11/θ
n
2s/2s1
.
6.79
20
International Journal of Mathematics and Mathematical Sciences
s
s
M ⊆ B2,∞
M, we have
For r ≥ 1 and p ≥ 2, since Bp,r
S4,2
j
∞ 2
−1
2
≤
βj,k
≤ C2−2j2 s ≤ C
jj2 1 k0
ln n11/θ
n
2s/2s1
6.80
.
2
s
For r ≥ 1, p ∈ 1, 2 and s > 1/p, using βj,k
{|βj,k|<2κλn} ≤ Cλn |βj,k |p , Bp,r
M ⊆ B2,∞
and 2s 12 − p/2 s 1/2 − 1/pp 2s, we have
2−p
S4,2 ≤
j
j1 2
−1
βj,k p C
2−p
Cλn
jj2 1 k0
≤C
≤C
ln n11/θ
n
ln n11/θ
n
2−p/2
2−p/2
ln n11/θ
n
∞
s1/2−1/p
M
j
j1 2
−1
βj,k p
jj2 1 k0
2−js1/2−1/pp
6.81
jj2 1
2−p/2
2
−j2 s1/2−1/pp
≤C
ln n11/θ
n
2s/2s1
.
So, for r ≥ 1, {p ≥ 2 and s > 0} or {p ∈ 1, 2 and s > 1/p}, we have
S4 ≤ C
2s/2s1
ln n11/θ
n
.
6.82
.
6.83
It follows from 6.61, 6.66, 6.75, and 6.82 that
S≤C
ln n11/θ
n
2s/2s1
Combining 6.55, 6.57, 6.60, and 6.83, we have, for r ≥ 1, {p ≥ 2 and s > 0} or
{p ∈ 1, 2 and s > 1/p},
1
2
fH x − fx dx
0
≤C
ln n11/θ
n
2s/2s1
The proof of Theorem 5.2 is complete.
Acknowledgment
This paper is supported by ANR Grant NatImages, ANR-08-EMER-009.
.
6.84
International Journal of Mathematics and Mathematical Sciences
21
References
1 S. T. Buckland, D. R. Anderson, K. P. Burnham, and J. L. Laake, Distance Sampling: Estimating
Abundance of Biological Populations, Chapman & Hall, London, UK, 1993.
2 D. Cox, “Some sampling problems in technology,” in New Developments in Survey Sampling, N. L.
Johnson and H. Smith Jr., Eds., pp. 506–527, John Wiley & Sons, New York, NY, USA, 1969.
3 J. Heckman, “Selection bias and self-selection,” in The New Palgrave : A Dictionary of Economics, pp.
287–296, MacMillan Press, New York, NY, USA, 1985.
4 G. P. Patil and C. R. Rao, “The weighted distributions: a survey of their applications,” in Applications
of Statistics, P. R. Krishnaiah, Ed., pp. 383–405, North-Holland, Amsterdam, The Netherlands, 1977.
5 H. El Barmi and J. S. Simonoff, “Transformation-based density estimation for weighted distributions,”
Journal of Nonparametric Statistics, vol. 12, no. 6, pp. 861–878, 2000.
6 S. Efromovich, “Density estimation for biased data,” The Annals of Statistics, vol. 32, no. 3, pp. 1137–
1161, 2004.
7 E. Brunel, F. Comte, and A. Guilloux, “Nonparametric density estimation in presence of bias and
censoring,” Test, vol. 18, no. 1, pp. 166–194, 2009.
8 C. Chesneau, “Wavelet block thresholding for density estimation in the presence of bias,” Journal of
the Korean Statistical Society, vol. 39, no. 1, pp. 43–53, 2010.
9 P. Ramı́rez and B. Vidakovic, “Wavelet density estimation for stratified size-biased sample,” Journal
of Statistical Planning and Inference, vol. 140, no. 22, pp. 419–432, 2010.
10 H. Doosti and I. Dewan, “Wavelet linear density estimation for associated stratified size-biased
sample,” Statistics & Mathematics Unit. In press.
11 P. Doukhan, Mixing. Properties and Examples, vol. 85 of Lecture Notes in Statistics, Springer, New York,
NY, USA, 1994.
12 M. Carrasco and X. Chen, “Mixing and moment properties of various GARCH and stochastic
volatility models,” Econometric Theory, vol. 18, no. 1, pp. 17–39, 2002.
13 A. Antoniadis, “Wavelets in statistics: a review with discussion,” Journal of the Italian Statistical
Society, Series B, vol. 6, pp. 97–144, 1997.
14 W. Härdle, G. Kerkyacharian, D. Picard, and A. Tsybakov, Wavelets, Approximation, and Statistical
Applications, vol. 129 of Lecture Notes in Statistics, Springer, New York, NY, USA, 1998.
15 D. L. Donoho, I. M. Johnstone, G. Kerkyacharian, and D. Picard, “Density estimation by wavelet
thresholding,” The Annals of Statistics, vol. 24, no. 2, pp. 508–539, 1996.
16 C. S. Withers, “Conditions for linear processes to be strong-mixing,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 57, no. 4, pp. 477–480, 1981.
17 D. S. Modha and E. Masry, “Minimum complexity regression estimation with weakly dependent
observations,” IEEE Transactions on Information Theory, vol. 42, no. 6, part 2, pp. 2133–2145, 1996.
18 A. Cohen, I. Daubechies, and P. Vial, “Wavelets on the interval and fast wavelet transforms,” Applied
and Computational Harmonic Analysis, vol. 1, no. 1, pp. 54–81, 1993.
19 Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics, Cambridge
University Press, Cambridge, UK, 1992.
20 Y.P. Chaubey, C. Chesneau, and H. Doosti, “On linear wavelet density estimation: some recent
developments,” Journal of the Indian Society of Agricultural Statistics. In press.
21 F. Leblanc, “Wavelet linear density estimator for a discrete-time stochastic process: Lp -losses,”
Statistics & Probability Letters, vol. 27, no. 1, pp. 71–84, 1996.
22 E. Masry, “Probability density estimation from dependent observations using wavelets orthonormal
bases,” Statistics & Probability Letters, vol. 21, no. 3, pp. 181–194, 1994.
23 B. Delyon and A. Juditsky, “On minimax wavelet estimators,” Applied and Computational Harmonic
Analysis, vol. 3, no. 3, pp. 215–228, 1996.
24 J. A. Davydov, “The invariance principle for stationary processes,” Theory of Probability and Its
Applications, vol. 15, pp. 498–509, 1970.
25 E. Rio, “The functional law of the iterated logarithm for stationary strongly mixing sequences,” The
Annals of Probability, vol. 23, no. 3, pp. 1188–1203, 1995.
26 E. Liebscher, “Strong convergence of sums of a-mixing random variables with applications to density
estimation,” Stochastic Processes and their Applications, vol. 65, no. 1, pp. 69–80, 1996.
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