# Optimal Wavelet Estimation of Density Derivatives for Size-Biased Data

Authors: Jinru Wang; Zijuan Geng; Fengfeng Jin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

A perfect achievement has been made for wavelet density estimation by Dohono et al. in 1996, when the samples without any noise are independent and identically distributed (i.i.d.). But in many practical applications, the random samples always have noises, and estimation of the density derivatives is very important for detecting possible bumps in the associated density. Motivated by Dohono's work, we propose new linear and nonlinear wavelet estimatorsf
^
lin
(
m
)
,
f
^
non
(
m
) for density derivatives f
(
m
) when the random samples have size-bias. It turns out that the linear estimation E
(
∥
f
^
lin
(
m
)
-
f
(
m
)
∥
p
) for f
(
m
)
∈
B
r
,
q
s
(
A
,
L
) attains the optimal covergence rate when r
≥
p, and the nonlinear one E
(
∥
f
^
lin
(
m
)
-
f
(
m
)
∥
p
) does the same if r
<
p.


---

## Body

## 1. Introduction

Wavelet analysis plays important roles in both pure and applied mathematics such as signal processing, image compress, numerical solution, and local fractional calculus [1, 2]. One of which is to estimate an unknown density function based on random samples [3–8]. The perfect achievement was made by Dohono et al. [9], when the i.i.d. samples X
1
,
X
2
,
…
,
X
n have not any noise. On the other hand, Besov spaces contain many functional spaces (e.g., Hölder spaces and Sobolev spaces with noninteger exponents) as their special examples. In some statistical models, the error is measured in L
p norm [9–13].In practice, it usually happens that getting the direct sample from a random variable is impossible. In this paper, we want to consider the true density functionf
X
(
x
). But we can only observe the samples Y
i
,
i
=
1,2
,
…
,
n, for the size-biased data; that is,
(1)
f
Y
(
y
)
=
g
(
y
)
f
X
(
y
)
μ
,
where g
(
y
) is the so-called bias function, μ
=
E
(
g
(
X
)
)
<
∞.In many cases, a linearg is recommended, but, in general, the form of g should be studied via additional experiments. The purpose of this paper is to estimate the derivatives of the true density functions f
X
(
m
), m
=
0,1
,
2
,
…; we study the optimal convergence rate of wavelet estimators in L
p norm over Besov spaces.Size-biased data arise when an observation depends on samples magnitude. Several examples of model (1) can be found in the literature [14]. For instance, in [15], it is shown that the distribution of the concentration of alcohol in the blood of intoxicated drivers is of interest; since the drunken drivers have a larger chance of being arrested, the collected data are size-biased.The estimation problem for biased data (1) has been discussed in some papers. In 1982, Vardi [16] considered the nonparametric maximum likelihood estimation for f
X
(
x
). In 1991, Jones [17] discussed the mean squared error properties of the kernel density estimation. In 2004, Efromovich [18] developed the Efromovich-Pinsker adaptive Fourier estimator. It was based on a block shrinkage algorithm and achieved the minimax rate of convergence under the L
2 risk over the Besov class B
2,2
s.In 2010, Ramírez and Vidakovic [14] proposed a linear wavelet estimator and discussed the consistency of function in L
2
[
0,1
] under the mean integrated squared error (MISE) sense. But the wavelet estimator in paper [14] contained the unknown parameter μ. In the same year, Chesneau [10] constructed a nonlinear wavelet estimator and evaluated the L
p risk in the Besov space B
r
,
q
s. But about the estimation of the density derivatives about model (1), to our knowledge, we have not seen any result. Estimation of the derivatives of a density is very important in detecting possible bumps.The current paper is organized as follows. In Section2, we briefly describe the preliminaries on wavelets and Besov space. The linear estimator and its convergence rate are presented in Section 3. In order to discuss optimality, Section 4 is devoted to give the lower bound for an arbitrary estimator. In Section 5, we consider nonlinear wavelet estimator and its optimal convergence rate. Our estimations improve the theorems in [10, 13, 14, 18].


## 2. Wavelets and Besov Spaces

In this section, we will recall some useful and well-known concepts and lemmas.In order to construct a wavelet basis, we need a structure inL
2
(
ℝ
) which can decompose L
2
(
ℝ
) into a direct sum of mutually orthogonal spaces.Definition 1 (see [19]).
A multiresolution analysis (MRA) ofL
2
(
ℝ
) is a set of increasing, closed linear subspaces V
j
⊂
V
j
+
1, for all j
∈
ℤ, called scaling spaces, satisfying(a)
⋂
-
∞
∞
V
j
=
{
0
}, ⋃
-
∞
∞
V
j
¯
=
L
2
(
ℝ
);
(b)
f
(
·
)
∈
V
0 if and only if f
(
2
j
·
)
∈
V
j all j
∈
ℤ;
(c)
f
(
·
)
∈
V
0 if and only if f
(
·
-
k
)
∈
V
0 for all k
∈
ℤ;
(d)
there exists a functionϕ
(
·
)
∈
V
0 such that {
ϕ
(
·
-
k
)
} is an orthogonal basis in V
0. The function ϕ
(
·
) is called the scaling function of the multiresolution analysis.With the standard notationf in wavelet analysis, there exists a corresponding wavelet function ψ
(
x
)
=
∑
k
(
-
1
)
k
h
1
-
k
¯
φ
1
k
(
x
)
,
h
k
=
〈
φ
,
φ
1
k
〉, such that for fixed j
,
{
ψ
j
,
k
(
t
)
}
k
∈
ℤ is an orthonormal basis of W
j which is the orthogonal complement of the space V
j in V
j
+
1
(
V
j
+
1
=
V
j
⊕
W
j
). For fixed J
∈
ℕ, both {
φ
J
k
(
x
)
,
ψ
j
,
k
(
x
)
}
j
≥
J
,
k
∈
ℤ and {
ψ
j
,
k
(
x
)
}
j
,
k
∈
ℤ are orthonormal bases of L
2
(
ℝ
).As usual,L
p
(
ℝ
)
(
p
≥
1
) denotes the classical Lebesgue space on the real line ℝ. Although wavelet bases are constructed for L
2
(
ℝ
), most of them constitute unconditional bases for L
p
(
ℝ
).Lemma 2 (see [20]).
Letφ be a compactly supported, orthonormal scaling function and ψ the corresponding wavelet. Then for any f
(
x
)
∈
L
p
(
ℝ
) with 1
≤
p
≤
∞, the following expansion:
(2)
∑
k
∈
ℤ
‍
α
J
,
k
φ
J
,
k
(
x
)
+
∑
j
≥
J
‍
∑
k
∈
ℤ
‍
β
j
,
k
ψ
j
,
k
(
x
)
converges to f
(
x
) for almost everywhere x
∈
ℝ, where
(3)
α
J
,
k
=
∫
ℝ
‍
f
(
x
)
φ
J
,
k
(
x
)
d
x
,
β
j
,
k
=
∫
ℝ
‍
f
(
x
)
ψ
j
,
k
(
x
)
d
x
.Lemma 3 (see [3]).
If the scaling functionφ satisfies ess
sup
∑
k
∈
ℤ
|
φ
(
x
-
k
)
|
<
∞, then for any sequence {
λ
k
}
k
∈
ℤ
∈
l
p, one has
(4)
C
1
∥
λ
∥
l
p
2
(
j
/
2
-
j
/
p
)
≤
∥
∑
k
‍
λ
k
φ
j
,
k
∥
p
≤
C
2
∥
λ
∥
l
p
2
(
j
/
2
-
j
/
p
)
,
where C
1
=
(
∥
θ
φ
∥
∞
1
/
p
∥
φ
∥
1
1
/
q
)
-
1, C
2
=
(
∥
θ
φ
∥
∞
1
/
q
∥
φ
∥
1
1
/
p
)
-
1, 1
≤
p
≤
∞, 1
/
p
+
1
/
q
=
1.Letting1
≤
p, q
≤
∞, s
:
=
n
+
α, α
∈
(
0,1
], the Besov spaces B
p
q
s
(
ℝ
) are defined by
(5)
B
p
q
s
(
ℝ
)
:
=
{
f
∣
f
∈
L
p
(
ℝ
)
,
f
(
n
)
∈
L
p
(
ℝ
)
,
ω
p
2
(
f
n
,
t
)
=
ε
(
t
)
t
α
,
∥
ε
∥
q
*
<
∞
}
with the associated norm ∥
f
∥
B
p
q
s
:
=
∥
f
∥
p
+
∥
f
(
n
)
∥
p
+
∥
ω
p
2
(
f
n
,
t
)
/
t
α
∥
q
*, where ω
p
2
(
f
,
t
)
:
=
sup
|
h
|
≤
t
∥
f
(
x
+
2
h
)
-
2
f
(
x
+
h
)
+
f
(
x
)
∥
p denotes the smoothness modulus of f, and ∥
ε
∥
q
*
:
=
{
(
∫
0
∞
‍
|
ε
(
t
)
|
q
(
d
t
/
t
)
)
1
/
q
,
1
≤
q
<
∞
,
ess
sup
t
|
ε
(
t
)
|
,
q
=
∞
.Between the different Besov spaces, the following embedding conclusions are established [3]. Let s
>
0, 1
≤
p
,
q
,
r
≤
∞; then(i)
B
p
q
s
↪
B
p
∞
s
↪
B
∞
∞
s
-
1
/
p, s
>
1
/
p;
(ii)
B
r
q
s
↪
B
p
q
s
′, r
<
p, s
′
=
s
-
1
/
r
+
1
/
p,whereA
↪
B denotes that the Banach space A is continuously embedding in the Banach space B; that is, there exists a constant c
>
0 such that, for any u
∈
A, we have ∥
u
∥
B
≤
c
∥
u
∥
A.A scaling functionφ
(
x
) is called t-regular, if φ
(
x
) has continuous derivatives of order t, and its corresponding wavelet ψ
(
x
) has vanishing moments of order t; that is,
(6)
∫
‍
x
k
ψ
(
x
)
d
x
=
0
,
k
=
0,1
,
…
,
t
-
1
.One of advantages of wavelets is that they can characterize Besov spaces.Lemma 4 (see [3]).
Letφ
(
x
) be a compact supported, t-regular orthonormal scaling function with the corresponding wavelet ψ
(
x
) and 0
<
s
<
t. If f
∈
L
p
(
ℝ
), α
0
k
:
=
〈
f
,
φ
0
k
〉
,
β
j
k
:
=
〈
f
,
ψ
j
k
〉, and 1
≤
p, q
≤
∞, then the following are equivalent:(i)
f
∈
B
p
q
s
(
ℝ
);
(ii)
(
2
j
s
∥
P
j
f
-
f
∥
p
)
j
≥
0
∈
l
q, where P
j is the projection operator to V
j; that is, P
j
f
:
=
∑
k
〈
f
,
φ
j
,
k
〉
φ
j
k;
(iii)
∥
α
0
·
∥
p
+
∥
{
2
j
(
s
+
1
/
2
-
1
/
p
)
∥
β
j
·
∥
p
}
j
≥
0
∥
q
<
+
∞. In this case,
(7)
∥
f
∥
B
p
q
s
~
∥
α
0
·
∥
p
+
∥
{
2
j
(
s
+
1
/
2
-
1
/
p
)
∥
β
j
·
∥
p
}
j
≥
0
∥
q
.Note 1.
The notationA
≲
B indicates that A
⩽
c
B with a positive constant c, which is independent of A and B. If A
≲
B and B
≲
A, we write A
~
B.In this paper, the Besov ballsB
~
r
q
s
(
A
,
L
) are defined by
(8)
B
~
r
q
s
(
A
,
L
)
:
=
{
f
∣
f
∈
B
r
q
s
(
ℝ
)
,
∥
f
∥
B
r
q
s
≤
L
,
|
supp
f
|
≤
A
}
.


## 3. Linear Estimator

In this section, we will give a linear estimator for density derivativesf
X
(
d
) in Besov spaces B
~
r
q
s
(
A
,
L
).The linear wavelet estimator of the derivative of a densityf
X
(
y
) is defined as follows:
(9)
f
^
X
,
lin
(
d
)
(
y
)
=
∑
k
‍
α
^
J
,
k
ϕ
J
,
k
(
y
)
,
where
(10)
α
^
J
,
k
:
=
(
-
1
)
d
μ
^
n
∑
i
=
1
n
‍
ϕ
J
,
k
(
Y
i
)
g
(
Y
i
)
,
μ
^
:
=
n
∑
i
=
1
n
(
1
/
g
(
Y
i
)
)
.The following inequalities play important roles in this paper.Lemma 5 (see [3] (Rosenthal inequality)).
LetX
1
,
…
,
X
n be independent random variables such that E
(
X
i
)
=
0 and |
X
i
|
<
M; then there exists a constant C
(
p
)
>
0 such that(i)
E
(
|
∑
i
=
1
n
X
i
|
p
)
≤
C
(
p
)
(
M
p
-
2
∑
i
=
1
n
E
(
|
X
i
|
2
)
+
(
∑
i
=
1
n
E
(
X
i
2
)
)
p
/
2
), p
>
2,
(ii)
E
(
|
∑
i
=
1
n
X
i
|
p
)
≤
C
(
p
)
(
∑
i
=
1
n
E
(
X
i
2
)
)
p
/
2, 0
<
p
≤
2.About the defined coefficients in (9), although E
(
α
^
J
,
k
)
≠
α
J
,
k, we have the following estimation.Lemma 6.
If2
j
≤
n, then, for any 1
≤
p
<
∞, one has E
|
α
^
J
k
-
α
J
k
|
p
≲
n
-
p
/
2
2
J
d
p.Proof.
By the definitions ofα
^
J
k, α
J
k and triangular inequality, one observes that
(11)
|
α
^
J
k
-
α
J
k
|
=
|
(
-
1
)
d
μ
^
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
=
|
μ
^
μ
(
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
)
+
μ
^
α
J
k
(
1
μ
-
1
μ
^
)
|
≤
|
μ
^
μ
|
·
|
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
+
|
μ
^
α
J
k
|
·
|
1
μ
^
-
1
μ
|
.
Since g
1
≤
g
(
y
)
≤
g
2 for any y
∈
ℝ, one has that
(12)
μ
^
=
n
∑
i
=
1
n
(
1
/
g
(
Y
i
)
)
≤
g
2
,
μ
=
E
g
(
X
)
≥
g
1
,
and α
J
k
=
∫
A
‍
f
X
(
d
)
(
y
)
ϕ
J
k
(
y
)
d
y. Thanks to embedding theorem B
r
∞
s
↪
1
B
∞
∞
s
-
1
/
r, for any s
>
1
/
r, one gets ∥
f
∥
∞
≤
∥
f
∥
B
∞
∞
s
-
1
/
r
≤
∥
f
∥
B
r
∞
s. It is easy to see |
μ
^
/
μ
|, |
μ
^
|, |
α
J
k
| are bounded. Using the convexity inequality, one obtains
(13)
E
|
α
^
J
k
-
α
J
k
|
p
≲
E
(
|
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
+
|
1
μ
^
-
1
μ
|
)
p
≤
2
p
-
1
E
(
|
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
p
+
|
1
μ
^
-
1
μ
|
p
)
≲
E
|
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
p
+
E
|
1
μ
^
-
1
μ
|
p
=
:
T
1
+
T
2
,
where
(14)
T
1
:
=
E
|
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
p
,
T
2
:
=
E
|
1
μ
^
-
1
μ
|
p
.
(i)
To estimateT
1:
(15)
T
1
=
:
E
|
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
p
=
E
|
1
n
∑
i
=
1
n
‍
(
(
-
1
)
d
μ
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
)
|
p
,denote ξ
i
:
=
(
-
1
)
d
μ
(
ϕ
J
k
(
d
)
(
Y
i
)
/
g
(
Y
i
)
)
-
α
J
k. Note that {
ξ
i
} are i.i.d. samples, and E
ξ
i
=
0. Moreover, for any given integer m
≥
2, one has
(16)
E
|
(
-
1
)
d
μ
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
|
m
=
E
|
μ
g
(
Y
i
)
(
ϕ
J
k
(
d
)
(
Y
i
)
)
2
|
·
|
(
μ
g
(
Y
i
)
)
m
-
1
(
ϕ
J
k
(
d
)
(
Y
i
)
)
m
-
2
|
≤
(
g
2
g
1
)
m
-
1
2
(
J
/
2
+
d
J
)
(
m
-
2
)
∥
ϕ
(
d
)
∥
∞
m
-
2
E
|
μ
g
(
Y
i
)
(
ϕ
J
k
(
d
)
(
Y
i
)
)
2
|
≲
2
(
J
/
2
+
d
J
)
(
m
-
2
)
E
|
μ
g
(
Y
i
)
(
ϕ
J
k
(
d
)
(
Y
i
)
)
2
|
,
E
|
μ
g
(
Y
i
)
(
ϕ
J
k
(
d
)
(
Y
i
)
)
2
|
=
∫
A
‍
μ
g
(
y
)
(
ϕ
J
k
(
d
)
(
y
)
)
2
f
Y
(
y
)
d
y
=
∫
A
‍
μ
g
(
y
)
(
ϕ
J
k
(
d
)
(
y
)
)
2
g
(
y
)
f
X
(
y
)
μ
d
y
=
∫
A
‍
f
X
(
y
)
(
ϕ
J
k
(
d
)
(
y
)
)
2
d
y
=
∫
A
‍
f
X
(
y
)
2
J
(
1
/
2
+
d
)
2
(
ϕ
J
k
(
d
)
(
2
J
y
-
k
)
)
2
d
y
≲
2
2
d
J
.
Therefore,
(17)
E
|
ξ
i
|
m
=
E
|
(
-
1
)
d
μ
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
α
J
k
|
m
≤
2
m
-
1
(
E
|
(
-
1
)
d
μ
ϕ
J
k
(
d
)
(
Y
i
)
g
(
Y
i
)
|
m
+
|
α
J
k
|
m
)
≲
2
J
(
1
/
2
+
d
)
m
-
J
.
By Rosenthal’s inequality Lemma 5,(
a
)
if1
⩽
p
⩽
2, one has
(18)
T
1
=
E
|
1
n
∑
i
=
1
n
‍
ξ
i
|
p
≤
n
-
p
(
∑
i
=
1
n
‍
E
|
ξ
i
|
2
)
p
/
2
=
n
-
p
(
n
E
|
ξ
i
|
2
)
p
/
2
≲
n
-
p
/
2
2
J
d
p
;
(
b
)
ifp
≥
2, one gets
(19)
T
1
=
E
|
1
n
∑
i
=
1
n
‍
ξ
i
|
p
≤
n
-
p
C
[
∑
i
=
1
n
‍
E
|
ξ
i
|
p
+
(
∑
i
=
1
n
‍
E
|
ξ
i
|
2
)
p
/
2
]
≲
n
-
p
(
n
2
J
(
1
/
2
+
d
)
p
-
J
)
+
n
-
p
(
∑
i
=
1
n
‍
E
|
ξ
i
|
2
)
p
/
2
≲
n
-
p
/
2
2
J
d
p
.
(ii)
To estimate the termT
2, since
(20)
T
2
=
:
E
|
1
μ
^
-
1
μ
|
p
=
E
|
1
n
∑
i
=
1
n
‍
1
g
(
Y
i
)
-
1
μ
|
p
=
E
|
1
n
∑
i
=
1
n
‍
(
1
g
(
Y
i
)
-
1
μ
)
|
p
,let η
i
:
=
1
/
g
(
Y
i
)
-
1
/
μ. It is easy to see E
η
i
=
0, and, for any integer m
≥
2, one obtains
(21)
E
|
η
i
|
m
=
E
|
1
g
(
Y
i
)
-
1
μ
|
m
≤
2
m
-
1
(
E
|
1
g
(
Y
i
)
|
m
+
μ
-
m
)
≤
2
m
-
1
(
g
1
-
m
+
g
1
m
)
:
=
C
(
m
)
.
Similarly, by Rosenthal’s inequality Lemma 5, (a)
for1
≤
p
<
2, m
=
2, one gets
(22)
T
2
=
E
|
1
n
∑
i
=
1
n
‍
η
i
|
p
≤
n
-
p
(
∑
i
=
1
n
‍
E
|
η
i
|
2
)
p
/
2
≲
n
-
p
·
n
p
/
2
=
n
-
p
/
2
;
(b)
forp
≥
2, that is, 1
-
p
≤
-
p
/
2, and m
=
p, one has
(23)
T
2
=
E
|
1
n
∑
i
=
1
n
‍
η
i
|
p
≤
n
-
p
C
(
p
)
(
∑
i
=
1
n
‍
E
|
η
i
|
p
+
(
∑
i
=
1
n
‍
E
|
η
i
|
2
)
p
/
2
)
≲
n
-
p
+
1
+
n
-
p
/
2
≲
n
-
p
/
2
.Summarizing the above estimation about T
1, T
2, one obtains that E
|
α
^
J
k
-
α
J
k
|
p
≲
n
-
p
/
2
2
J
d
p.Theorem 7.
Let scaling functionϕ
(
x
) be compactly supported and t-regular. If f
^
X
,
l
i
n
(
d
) be the estimator defined in (9), then for 1
≤
r
,
p
,
q
<
∞, 1
/
r
<
s
<
t
-
d, one has
(24)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
X
,
l
i
n
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≲
n
-
s
′
p
/
(
1
+
2
(
s
′
+
d
)
)
,
where s
′
=
s
-
(
1
/
r
-
1
/
p
)
+.Proof.
Firstly, using triangular inequality and convexity inequality, we decomposeE
∥
f
^
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p into the bias term and the stochastic term; that is,
(25)
E
∥
f
^
X
,
lin
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≤
E
(
∥
f
^
X
,
lin
(
d
)
(
y
)
-
P
J
f
X
(
d
)
(
y
)
∥
p
+
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
)
p
≤
2
p
-
1
(
E
∥
f
^
X
,
lin
(
d
)
(
y
)
-
P
J
f
X
(
d
)
(
y
)
∥
p
+
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
)
.
For the bias term ∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p, one can estimate it as follows.(i)
Whenr
=
p, Lemma 4 reduces to
(26)
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≲
2
-
J
s
p
.
(
ii
)
Whenr
<
p, using Besov space embedding theorems B
r
q
s
↪
B
p
q
s
′
,
s
′
=
s
-
1
/
r
+
1
/
p, one has
(27)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≤
sup
f
X
(
d
)
∈
B
~
p
q
s
′
(
A
,
L
)
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≲
2
-
J
s
′
p
.
(
iii
)
Whenr
>
p,  Hölder’s inequality and Lemma 4 tell us that
(28)
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
=
∫
A
‍
|
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
|
p
d
y
⩽
(
∫
A
‍
|
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
|
p
·
(
r
/
p
)
d
y
)
p
/
r
×
(
∫
A
‍
1
d
y
)
1
-
p
/
r
≲
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
r
p
≲
2
-
J
s
p
.Hence, for 1
≤
r
<
∞, one obtains that
(29)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
∥
P
J
f
X
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≲
2
-
J
s
′
p
.
Next, we estimate the stochastic term E
∥
f
^
X
(
d
)
(
y
)
-
P
J
f
X
(
d
)
(
y
)
∥
p
p. Clearly, due to Lemmas 3 and 6, one gets
(30)
E
∥
f
^
X
,
lin
(
d
)
(
y
)
-
P
J
f
X
(
d
)
(
y
)
∥
p
p
=
E
∥
∑
k
‍
(
α
^
J
k
-
α
J
k
)
ϕ
J
,
k
(
y
)
∥
p
p
≲
2
(
J
/
2
-
J
/
p
)
p
∑
k
‍
E
|
(
α
^
J
k
-
α
J
k
)
|
p
≲
n
-
p
/
2
2
J
(
1
/
2
+
d
)
p
.
By choosing J such that 2
J
(
n
)
~
n
1
/
(
1
+
2
(
s
′
+
d
)
), one obtains that
(31)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
X
,
lin
(
d
)
(
y
)
-
f
X
(
d
)
(
y
)
∥
p
p
≲
2
-
J
s
′
p
+
n
-
p
/
2
2
J
(
1
/
2
+
d
)
p
≲
n
-
s
′
p
/
(
1
+
2
(
s
′
+
d
)
)
.Remark 8.
Theorem7 can be considered as natural extension of [14] if d
=
0, p
=
r
=
2. Moreover, the next part shows the optimality of our linear estimation for r
⩾
p.


## 4. Lower Bound and Optimality

This section is devoted to showing that the linear estimator defined in (9) attains the optimal covergence rate  for r
≥
p. The idea of proof is motivated by [21].Lemma 9 (Varshamov-Gilbert Lemma [5]).
LetΘ
:
=
{
ε
=
(
ε
1
,
…
,
ε
m
)
}, ε
i
∈
{
0,1
}; then there exists a subset {
ε
0
,
…
,
ε
M
} of Θ with ε
0
=
(
0
,
…
,
0
) such that M
≥
2
m
-
3 and ∑
k
=
1
m
|
ε
k
i
-
ε
k
j
|
≥
m
/
8,0
≤
i
≠
j
≤
M.Lemma 10 (Fano’s lemma [22]).
Let(
Ω
,
ℵ
,
P
k
) be probability measurable spaces and A
k
∈
ℵ, k
=
0,1
,
…
,
m. If A
k
∩
A
v
=
ϕ for ∀
k
≠
v, one has
(32)
sup
0
≤
k
≤
m
P
k
(
A
k
c
)
≥
min
{
1
2
,
m
exp
(
-
3
e
-
1
-
κ
m
)
}
,
where A
k
c stands for the complement of A
k
,
κ
m
:
=
inf
0
≤
v
≤
m
(
1
/
m
)
∑
k
≠
v
K
(
P
k
,
P
v
) and K
(
P
k
,
P
v
) stands for Kullback distance in [5].Based on the above lemmas, we have the following lower bound estimation.Theorem 11.
Letf
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
) with 1
≤
r, q
≤
∞, 1
≤
p
<
∞ and s
>
1
/
r; there exist two constants g
1 and g
2 such that 0
<
g
1
≤
g
(
x
)
≤
g
2
<
∞. If f
^
X
(
d
) is any estimator of f
X
(
d
) with i.i.d. random samples, then(33)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
n
(
d
)
-
f
X
(
d
)
∥
p
≳
max
{
(
ln
n
n
)
(
s
-
1
/
r
+
1
/
p
)
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
n
-
s
/
(
2
(
s
+
d
)
+
1
)
}
=
{
(
ln
n
n
)
(
s
-
1
/
r
+
1
/
p
)
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
≤
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.Proof.
(i) Firstly, we prove(34)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
n
(
d
)
-
f
X
(
d
)
∥
p
≳
(
ln
n
n
)
(
s
-
1
/
r
+
1
/
p
)
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
.
It is sufficient to construct h
k
(
x
) such that h
k
(
d
)
∈
B
~
r
q
s
(
A
,
L
) and
(35)
sup
k
E
∥
f
^
n
(
d
)
-
h
k
(
d
)
∥
p
≳
(
ln
n
n
)
(
s
-
1
/
r
+
1
/
p
)
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
.
Suppose that φ is a compactly supported, t
(
t
>
s
+
d
) regular and orthonormal scaling function and ψ is the corresponding wavelet with supp
ψ
⊆
[
0
,
l
)
,
l
∈
N
+. Assume h
(
x
)
∈
B
~
r
q
s
+
d
(
A
,
L
/
2
)
,
h
(
x
)
∣
[
0
,
l
)
=
C
0
>
0. Define Δ
j
:
=
{
0
,
l
,
2
l
,
…
,
(
2
j
-
1
)
l
,
2
j
l
}, a
j
:
=
2
-
j
(
s
+
d
+
1
/
2
-
1
/
r
) and
(36)
h
k
(
x
)
:
=
h
(
x
)
+
a
j
ψ
j
k
(
x
)
I
{
k
≠
2
j
l
}
,
k
∈
Δ
j
.
Obviously, h
2
j
l
=
h
(
x
) and h
k
(
x
)
∈
B
~
r
q
s
+
d
(
A
,
L
); that is, h
k
(
d
)
(
x
)
∈
B
~
r
q
s
(
A
,
L
). Moreover, supp
ψ
j
k
∩
supp
j
k
′
=
∅ for k
,
k
′
∈
Δ
j and k
≠
k
′. So, one gets
(37)
∥
h
k
(
d
)
-
h
k
′
(
d
)
∥
p
=
∥
a
j
(
ψ
j
k
(
d
)
-
ψ
j
k
′
(
d
)
)
∥
p
≥
a
j
∥
ψ
j
k
(
d
)
∥
p
=
2
-
j
(
s
+
1
/
p
-
1
/
r
)
∥
ψ
(
d
)
∥
p
:
=
η
j
.
Clearly, A
k
:
=
{
∥
f
^
n
(
d
)
-
h
k
(
d
)
∥
p
<
η
j
/
2
} satisfies A
k
∩
A
k
′
=
∅ for k
≠
k
′. Then, Fano’s Lemma 10 tells us that
(38)
sup
k
∈
Δ
j
P
h
k
n
(
A
k
c
)
≥
min
{
1
2
,
2
j
exp
(
-
3
e
-
1
-
κ
2
j
)
}
.
On the other hand, one has
(39)
E
∥
f
^
n
(
d
)
(
x
)
-
h
k
(
d
)
(
x
)
∥
p
≥
η
j
2
P
h
Y
k
n
(
∥
f
^
n
(
d
)
-
h
k
(
d
)
∥
p
≥
η
j
2
)
=
η
j
2
P
h
Y
k
n
(
A
k
c
)
.
Then
(40)
sup
k
∈
Δ
j
E
(
∥
f
^
n
(
d
)
(
x
)
-
h
k
(
d
)
(
x
)
∥
p
)
≥
η
j
2
sup
k
∈
Δ
j
P
h
Y
k
n
(
A
k
c
)
≥
η
j
2
min
{
1
2
,
2
j
exp
(
-
3
e
-
1
-
κ
2
j
)
}
,
where κ
2
j
:
=
inf
k
∈
Δ
j
1
/
2
j
∑
k
≠
v
K
(
P
h
Y
k
n
,
P
h
Y
v
n
). Next, one shows that κ
2
j
≤
g
2
C
0
-
1
n
a
j
2
/
g
1.
Recall that(41)
K
(
P
1
n
,
P
2
n
)
≔
∫
f
1
n
·
f
2
n
>
0
‍
f
1
n
(
x
)
ln
f
1
n
(
x
)
f
2
n
(
x
)
d
x
=
∑
i
=
1
n
‍
∫
‍
f
1
(
x
i
)
ln
f
1
(
x
i
)
f
2
(
x
i
)
d
x
i
=
n
K
(
P
1
1
,
P
2
1
)
,
where f
i
n
(
x
)
:
=
f
i
(
x
1
)
f
i
(
x
2
)
⋯
f
i
(
x
n
), i
=
1,2. Note that, ln
u
≤
u
-
1, if u
>
0, one has
(42)
K
(
P
1
n
,
P
2
n
)
=
n
∫
‍
f
1
(
x
)
ln
f
1
(
x
)
f
2
(
x
)
d
x
≤
n
∫
‍
1
f
2
(
x
)
[
f
1
(
x
)
-
f
2
(
x
)
]
2
d
x
.
Since h
k
Y
(
x
)
=
g
(
x
)
h
k
(
x
)
/
μ and supp
|
h
k
(
x
)
-
h
(
x
)
|
⊂
[
0
,
l
),
(43)
κ
2
j
≔
inf
k
∈
Δ
j
2
-
j
∑
k
≠
v
‍
K
(
P
h
Y
k
n
,
P
h
Y
v
n
)
≤
2
-
j
∑
k
∈
Δ
j
,
k
≠
2
j
l
‍
K
(
P
h
Y
k
n
,
P
h
Y
n
)
≤
2
-
j
n
∑
k
∈
Δ
j
,
k
≠
2
j
l
∫
μ
g
(
x
)
h
(
x
)
‍
×
[
g
(
x
)
h
k
(
x
)
μ
-
g
(
x
)
h
(
x
)
μ
]
2
d
x
≤
g
2
g
1
2
-
j
n
∑
k
∈
Δ
j
,
k
≠
2
j
l
‍
C
0
-
1
∫
‍
a
j
2
|
ψ
j
k
|
2
d
x
=
g
2
g
1
C
0
-
1
n
a
j
2
.
Taking 2
j
~
(
n
/
ln
n
)
1
/
(
2
(
s
+
d
-
1
/
r
)
+
1
), then n
a
j
2
=
n
2
-
2
j
(
s
+
d
+
1
/
2
-
1
/
r
)
~
ln
n. One can choose C such that n
a
j
2
≤
C
ln
n and C
g
2
[
4
(
s
+
d
-
1
/
r
)
+
2
]
<
C
0
g
1. Therefore
(44)
2
j
e
-
κ
2
j
≥
2
j
e
-
g
2
C
0
-
1
n
a
j
2
/
g
1
≳
(
n
ln
n
)
[
4
(
s
+
d
-
1
/
r
)
+
2
]
-
1
n
-
g
2
C
0
-
1
C
/
g
1
≥
1
.
One has
(45)
sup
k
∈
Δ
j
E
(
∥
f
^
n
(
d
)
(
x
)
-
h
k
(
d
)
(
x
)
∥
p
)
≥
η
j
2
sup
k
∈
Δ
j
P
h
Y
k
n
(
A
k
c
)
≥
C
η
j
.
Noting that η
j
=
2
-
j
(
s
+
1
/
p
-
1
/
r
)
∥
ψ
(
d
)
∥
p, one gets
(46)
sup
k
∈
Δ
j
E
(
∥
f
^
n
(
d
)
(
x
)
-
h
k
(
d
)
(
x
)
∥
p
)
≳
(
ln
n
n
)
(
s
+
1
/
p
-
1
/
r
)
/
(
2
(
s
+
d
-
1
/
r
)
+
1
)
.
(ii)
Next, we prove(47)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
n
(
d
)
-
f
X
(
d
)
∥
p
≳
n
-
s
/
(
2
(
s
+
d
)
+
1
)
.Similarly, it is sufficient to construct h
ε
i
(
x
), i
=
0,1
,
…
,
M such that h
ε
i
(
d
)
(
x
)
∈
B
~
r
q
s
(
A
,
L
) and
(48)
sup
i
E
∥
f
^
n
(
d
)
(
x
)
-
h
ε
i
(
d
)
(
x
)
∥
p
≳
n
-
s
/
(
2
(
s
+
d
)
+
1
)
.
Similarly to prove (i), suppose that h
(
x
)
∈
B
~
r
q
s
+
d
(
A
,
L
/
2
), h
(
x
)
|
[
0
,
l
)
=
C
0
>
0. Defining a
~
j
:
=
2
-
j
(
s
+
d
+
1
/
2
), Δ
~
j
:
=
{
0
,
l
,
2
l
,
…
,
(
2
j
-
1
)
l
} and
(49)
h
ε
i
(
x
)
:
=
h
(
x
)
+
a
~
j
∑
k
∈
Δ
~
j
‍
ε
k
i
ψ
j
,
k
(
x
)
,
i
=
0,1
,
…
,
M
,
with ε
i
=
(
ε
k
i
)
k
∈
Δ
~
j
∈
{
0,1
}
2
j. Moreover, since ϵ
k
∈
{
0,1
}, one knows that ∑
k
∈
Δ
~
j
|
ε
k
i
|
r
≤
2
j and
(50)
2
j
(
s
+
d
+
1
/
2
-
1
/
r
)
a
~
j
(
∑
k
∈
Δ
~
j
‍
|
ε
k
i
|
r
)
1
/
r
≤
1
.
By Lemma 10, ∥
a
~
j
∑
k
∈
Δ
~
j
ε
k
i
ψ
j
k
(
x
)
∥
B
r
q
s
+
d
≤
C. Hence h
ε
i
(
d
)
(
x
)
∈
B
~
r
q
s
(
A
,
L
). According to Lemma 9, there exist {
ε
0
,
…
,
ε
M
} such that M
≥
2
2
j
-
3 and
(51)
∑
k
∈
Δ
~
j
‍
|
ε
k
l
-
ε
k
i
|
≥
2
j
8
=
2
j
-
3
,
0
≤
l
≠
i
≤
M
.
Since supp
ψ
j
k
∩
supp
ψ
j
k
′
=
∅ for k
≠
k
′
∈
Δ
~
j, this leads to ∥
h
ε
l
(
d
)
-
h
ε
i
(
d
)
∥
p
p
≥
2
-
j
s
p
·
2
-
3
∥
ψ
∥
p
p and
(52)
∥
h
ε
l
(
d
)
-
h
ε
i
(
d
)
∥
p
≥
2
-
j
s
·
8
-
1
/
p
∥
ψ
∥
p
:
=
η
j
.
Clearly, the sets A
ε
i
:
=
{
∥
f
^
n
(
d
)
-
h
ε
i
(
d
)
∥
p
<
η
j
/
2
}, i
=
0,1
,
…
,
M, satisfy A
ε
l
∩
A
ε
i
=
∅ for ∀
l
≠
i. Using Fano’s Lemma 10, one has
(53)
sup
0
≤
i
≤
M
P
h
ε
i
n
(
A
ε
i
c
)
≥
min
{
1
2
,
M
exp
(
-
3
e
-
1
-
κ
M
)
}
,
where κ
M
:
=
inf
0
≤
v
≤
M
(
1
/
M
)
∑
i
≠
v
K
(
P
h
Y
ε
i
n
,
P
h
Y
ε
v
n
), and one can get κ
M
≤
(
g
2
/
g
1
)
C
0
-
1
n
a
~
j
2
2
j due to the similar arguments as (i).
Taking2
j
~
n
1
/
(
2
(
s
+
d
)
+
1
), then n
a
~
j
2
=
n
2
-
2
j
(
s
+
d
+
1
/
2
)
~
1. One can choose a constant C
>
0 such that n
a
~
j
2
≤
C
<
(
g
1
/
32
g
2
)
C
0. By M
≥
2
2
j
-
3, then one obtains
(54)
M
e
-
κ
M
≥
2
(
1
/
2
)
(
2
j
-
3
)
e
-
g
2
C
0
-
1
n
a
~
j
2
2
j
g
1
-
1
≥
1
.
On the other hand, η
j
:
=
8
-
1
/
p
∥
ψ
∥
p
2
-
j
s
~
n
-
s
/
(
2
(
s
+
d
)
+
1
) reduces to
(55)
sup
0
≤
i
≤
M
E
(
∥
f
^
n
(
d
)
-
h
ε
i
(
d
)
∥
p
)
≥
sup
0
≤
i
≤
M
η
j
2
P
h
Y
ε
i
n
(
A
ε
i
c
)
≥
C
η
j
.
Therefore, one gets the following desired result:
(56)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
n
(
d
)
-
f
X
(
d
)
∥
p
≳
n
-
s
/
(
2
(
s
+
d
)
+
1
)
.Note thats
′
=
s, if r
≥
p. Then we have the following corollary.Corollary 12.
Ifr
≥
p, the linear estimator (9) attains the optimal covergence rate.


## 5. Nonlinear Estimator

In this paper, the nonlinear wavelet estimator is defined as follows:(57)
f
^
X
,
non
(
d
)
(
y
)
=
∑
k
‍
α
^
j
0
k
ϕ
j
0
k
(
y
)
+
∑
j
=
j
0
j
1
‍
∑
k
‍
β
^
j
k
*
ψ
j
k
(
y
)
,
where
(58)
α
^
j
0
k
:
=
(
-
1
)
d
μ
^
n
∑
i
=
1
n
‍
ϕ
j
0
k
(
d
)
(
Y
i
)
g
(
Y
i
)
,
β
^
j
k
:
=
(
-
1
)
d
μ
^
n
∑
i
=
1
n
‍
ψ
j
k
(
d
)
(
Y
i
)
g
(
Y
i
)
,
μ
^
:
=
n
∑
i
=
1
n
(
1
/
g
(
Y
i
)
)
.
The hard thresholding wavelet coefficients are β
^
j
k
*
:
=
β
^
j
k
I
{
|
β
^
j
k
|
≥
λ
}, where
(59)
I
{
|
β
^
j
k
|
≥
λ
}
:
=
{
1
,
|
β
^
j
k
|
≥
λ
,
0
,
|
β
^
j
k
|
<
λ
.
λ
=
c
2
j
d
j
n
.About the wavelet coefficients, we can get the following lemmas whose proof is very similar to Lemma6 and we omit it.Lemma 13.
If2
j
≤
n, then, for any 1
≤
p
<
∞, one has E
|
β
j
k
-
β
^
j
k
|
p
≲
n
-
p
/
2
·
2
j
d
p.Lemma 14 (see [3] (Bernstein inequality)).
LetX
1
,
X
2
,
…
,
X
n be independent random variables such that |
X
i
|
≤
M
<
∞
,
E
(
X
i
)
=
0, b
n
2
:
=
∑
i
=
1
n
E
(
X
i
2
); then
(60)
P
(
|
∑
i
=
1
n
‍
X
i
|
>
λ
)
≤
2
exp
(
-
λ
2
2
(
b
n
2
+
M
λ
/
3
)
)
,
∀
λ
>
0
.Lemma 15.
Ifj
2
j
≤
n, then, for any ω
>
0, there exists a constant c
>
0 such that
(61)
P
(
|
β
^
j
k
-
β
j
k
|
>
λ
=
c
2
j
d
j
n
)
≲
2
-
ω
j
.Proof.
One can easily get(62)
μ
^
≤
g
2
,
μ
≥
g
1
,
1
μ
≤
g
1
-
1
,
|
β
j
,
k
|
≤
A
1
/
2
∥
f
X
(
d
)
∥
∞
≤
A
1
/
2
∥
f
X
(
d
)
∥
B
r
q
s
,
|
β
^
j
,
k
-
β
j
,
k
|
=
|
μ
^
μ
(
(
-
1
)
d
μ
n
∑
i
=
1
n
‍
ψ
j
,
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
β
j
,
k
)
+
μ
^
β
j
,
k
(
1
μ
-
1
μ
^
)
|
≲
|
1
n
∑
i
=
1
n
‍
(
(
-
1
)
d
μ
ψ
j
,
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
β
j
,
k
)
|
+
|
1
n
∑
i
=
1
n
‍
(
1
g
(
Y
i
)
-
1
μ
)
|
=
:
|
1
n
∑
i
=
1
n
‍
ξ
i
|
+
|
1
n
∑
i
=
1
n
‍
η
i
|
,
where ξ
i
=
(
-
1
)
d
(
μ
ψ
j
,
k
(
d
)
(
Y
i
)
/
g
(
Y
i
)
)
-
β
j
,
k, η
i
=
1
/
g
(
Y
i
)
-
1
/
μ. So, one obtains that
(63)
P
(
|
β
^
j
,
k
-
β
j
,
k
|
>
λ
)
≤
P
(
|
1
n
∑
i
=
1
n
‍
ξ
i
|
+
|
1
n
∑
i
=
1
n
‍
η
i
|
>
λ
)
≤
P
(
|
1
n
∑
i
=
1
n
‍
ξ
i
|
>
λ
2
)
+
P
(
|
1
n
∑
i
=
1
n
‍
η
i
|
>
λ
2
)
.
Now, we estimate P
(
|
(
1
/
n
)
∑
i
=
1
n
ξ
i
|
>
λ
/
2
). Clearly, E
ξ
i
=
0 and E
ξ
i
2
≲
2
2
j
d
=
:
σ
1
2. Moreover, one has
(64)
|
ξ
i
|
=
|
(
-
1
)
d
μ
ψ
j
,
k
(
d
)
(
Y
i
)
g
(
Y
i
)
-
(
-
1
)
d
E
(
μ
ψ
j
,
k
(
d
)
(
Y
i
)
g
(
Y
i
)
)
|
≤
2
g
2
g
1
-
1
·
2
j
(
1
/
2
+
d
)
∥
ψ
(
d
)
∥
∞
=
:
M
1
2
j
(
1
/
2
+
d
)
,
where M
1
:
=
2
g
2
g
1
-
1
∥
ψ
(
d
)
∥
∞. By Bernstein’s inequality and λ
=
c
2
j
d
j
/
n
,
j
2
j
≤
n, one knows that
(65)
P
(
|
1
n
∑
i
=
1
n
‍
ξ
i
|
>
λ
2
)
≤
2
exp
(
-
n
(
λ
/
2
)
2
2
(
σ
1
2
+
M
1
·
2
j
(
1
/
2
+
d
)
λ
/
6
)
)
=
2
exp
(
-
n
c
2
2
2
j
d
(
j
/
n
)
/
4
2
(
2
2
j
d
+
M
1
·
2
j
(
1
/
2
+
d
)
c
2
j
d
j
/
n
/
6
)
)
=
2
exp
(
-
c
2
/
4
2
(
1
+
M
1
c
/
6
)
j
)
.
Taking c
1
>
0 such that (
c
2
/
4
)
/
2
(
1
+
M
1
c
/
6
)
≥
ω, then
(66)
P
(
|
1
n
∑
i
=
1
n
‍
ξ
i
|
>
λ
2
)
≤
2
e
-
ω
j
≲
2
-
ω
j
.
Next, we estimate P
(
|
(
1
/
n
)
∑
i
=
1
n
η
i
|
>
λ
/
2
). It is easy to know that
(67)
E
η
i
=
E
(
1
g
(
Y
i
)
)
-
E
(
1
μ
)
=
∫
‍
1
g
(
y
)
f
Y
(
y
)
d
y
-
1
μ
=
∫
‍
1
g
(
y
)
g
(
y
)
f
X
(
y
)
μ
d
y
-
1
μ
=
1
μ
∫
‍
f
X
(
y
)
d
y
-
1
μ
=
0
,
E
η
i
2
=
E
(
1
g
(
Y
i
)
-
1
μ
)
2
≤
2
(
E
|
1
g
(
Y
i
)
|
2
+
1
μ
2
)
≤
4
g
1
2
=
:
σ
2
2
,
|
η
i
|
=
|
1
g
(
Y
i
)
-
1
μ
|
=
|
1
g
(
Y
i
)
-
E
1
g
(
Y
i
)
|
≤
2
g
1
-
1
=
:
M
2
.
Since j
2
j
≤
n, Bernstein’s inequality tells us that
(68)
P
(
|
1
n
∑
i
=
1
n
‍
η
i
|
>
λ
2
)
≤
2
exp
(
-
n
(
λ
/
2
)
2
2
(
σ
2
2
+
M
2
λ
/
3
)
)
=
2
exp
(
-
n
c
2
2
2
j
d
(
j
/
n
)
/
4
2
(
σ
2
2
+
M
2
c
2
j
d
j
/
n
/
3
)
)
≤
2
exp
(
-
c
2
/
4
2
(
σ
2
2
+
M
2
c
/
3
)
j
)
.
Taking c
2
>
0 such that (
c
2
/
4
)
/
2
(
σ
2
2
+
M
2
c
/
3
)
≥
ω, one gets
(69)
P
(
|
1
n
∑
i
=
1
n
‍
η
i
|
>
λ
2
)
≤
2
e
-
ω
j
≲
2
-
ω
j
.
Letting c
=
max
{
c
1
,
c
2
}, by (66) and (69), one obtains that P
(
|
β
^
j
,
k
-
β
j
,
k
|
>
λ
)
≲
2
-
ω
j.Lemma 16.
Suppose that the parametersj
0, j
1, λ of the wavelet estimator defined in (57) satisfy the assumptions:
(70)
2
j
0
~
{
(
(
ln
n
)
(
p
-
r
)
/
r
n
)
1
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
n
(
1
-
2
/
p
)
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
≤
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(71)
2
j
1
~
{
n
s
/
s
′
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
n
ln
n
)
1
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
≤
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
;
then one has
(72)
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
c
4
n
-
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
where c
3, c
4 are positive constants.Proof.
According to Lemma3, one obtains that
(73)
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≤
E
∑
j
=
j
0
j
1
‍
∥
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≲
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
*
-
β
j
k
|
p
)
1
/
p
.
Recalling that β
^
j
k
*
=
β
^
j
k
I
{
|
β
^
j
k
|
>
λ
}, one has
(74)
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≲
E
(
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
|
p
)
1
/
p
×
(
I
{
|
β
^
j
k
|
>
λ
,
|
β
j
k
|
≥
λ
2
}
+
I
{
|
β
^
j
k
|
>
λ
,
|
β
j
k
|
<
λ
2
}
)
+
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
j
k
|
p
)
1
/
p
×
(
I
{
|
β
^
j
k
|
≤
λ
,
|
β
j
k
|
≤
2
λ
}
+
I
{
|
β
^
j
k
|
≤
λ
,
|
β
j
k
|
>
2
λ
}
)
(
∑
k
‍
β
^
j
k
-
β
j
k
|
p
)
1
/
p
)
.
Note that
(75)
I
{
|
β
^
j
k
|
>
λ
,
|
β
j
k
|
<
λ
2
}
≤
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
,
I
{
|
β
^
j
k
|
≤
λ
,
|
β
j
k
|
>
2
λ
}
≤
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
,
and one gets |
β
^
j
k
-
β
j
k
|
≥
|
β
j
k
|
-
|
β
^
j
k
|
>
|
β
j
k
|
/
2; that is, |
β
j
k
|
<
2
|
β
^
j
k
-
β
j
k
|, if |
β
^
j
k
|
≤
λ
,
|
β
j
k
|
>
2
λ. Therefore,
(76)
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≲
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
j
k
|
≥
λ
2
}
)
1
/
p
+
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
1
/
p
+
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
j
k
|
p
I
{
|
β
j
k
|
≤
2
λ
}
)
1
/
p
=
:
e
21
+
e
22
+
e
23
,
where
(77)
e
21
:
=
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
j
k
|
≥
λ
2
}
)
1
/
p
,
e
22
≔
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
×
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
1
/
p
,
e
23
:
=
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
j
k
|
p
I
{
|
β
j
k
|
≤
2
λ
}
)
1
/
p
.
(i)
Firstly, we estimate(78)
e
23
:
=
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
j
k
|
p
I
{
|
β
j
k
|
≤
2
λ
}
)
1
/
p
.Let ξ
:
=
(
1
/
2
)
(
(
r
/
p
)
(
2
s
+
1
+
2
d
)
-
(
1
+
2
d
)
). By I
{
|
β
j
k
|
≤
2
λ
}
≤
(
2
λ
/
|
β
j
k
|
)
p
-
r
(
r
<
p
), one obtains that
(79)
e
23
≲
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
j
k
|
p
(
2
λ
|
β
j
k
|
)
p
-
r
)
1
/
p
=
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
2
λ
)
(
p
-
r
)
/
p
(
∑
k
‍
|
β
j
k
|
r
)
1
/
p
=
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
2
λ
)
(
p
-
r
)
/
p
∥
β
j
·
∥
r
r
/
p
.
On the other hand, f
X
(
d
)
∈
B
~
r
q
s
(
ℝ
) implies ∥
β
j
·
∥
r
≲
2
-
j
(
s
+
1
/
2
-
1
/
r
). Taking λ
=
c
2
j
d
j
/
n
~
c
2
j
d
(
ln
n
)
/
n
,
j
1
-
j
0
~
ln
n, one has
(80)
e
23
≲
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
λ
(
p
-
r
)
/
p
2
-
j
(
s
+
1
/
2
-
1
/
r
)
(
r
/
p
)
≲
(
ln
n
n
)
(
p
-
r
)
/
2
p
∑
j
=
j
0
j
1
‍
2
-
j
(
1
/
2
)
[
(
r
/
p
)
(
2
s
+
1
+
2
d
)
-
(
1
+
2
d
)
]
=
(
ln
n
n
)
(
p
-
r
)
/
2
p
∑
j
=
j
0
j
1
‍
2
-
j
ξ
≲
(
ln
n
n
)
(
p
-
r
)
/
2
p
(
2
-
j
0
ξ
I
{
ξ
>
0
}
+
2
-
j
1
ξ
I
{
ξ
<
0
}
+
(
j
1
-
j
0
+
1
)
I
{
ξ
=
0
}
2
-
j
0
ξ
I
)
.
Note that ξ
>
0 if and only if r
>
(
1
+
2
d
)
p
/
(
2
(
s
+
d
)
+
1
). When ξ
=
0, that is, p
=
r
(
2
s
+
1
+
2
d
)
/
(
1
+
2
d
), one can compute s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
=
(
p
-
r
)
/
2
p. From (70), (71), one obtains that
(81)
e
23
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
c
4
n
-
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
where c
3, c
4 are positive constants.(ii)
We estimate(82)
e
21
:
=
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
j
k
|
≥
λ
2
}
)
1
/
p
.Due to Lemma 13, one has
(83)
E
|
β
^
j
k
-
β
j
k
|
p
≲
n
-
p
/
2
2
j
d
p
.
Using I
{
|
β
j
k
|
≥
λ
/
2
}
≤
(
|
β
j
k
|
/
(
λ
/
2
)
)
r and Jensen’s inequality, it can be proved that
(84)
e
21
=
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
j
k
|
≥
λ
2
}
)
1
/
p
≤
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
E
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
j
k
|
>
λ
2
}
)
1
/
p
≲
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
n
-
p
/
2
(
|
β
j
k
|
λ
/
2
)
r
)
1
/
p
=
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
n
-
1
/
2
2
j
d
λ
-
r
/
p
∥
β
j
·
∥
r
r
/
p
.
Similar to the proof of e
23, by ∥
β
j
·
∥
r
≲
2
-
j
(
s
+
1
/
2
-
1
/
r
) and λ
~
c
2
j
d
(
ln
n
)
/
n
,
j
1
-
j
0
~
ln
n, one gets
(85)
e
21
≲
∑
j
=
j
0
j
1
‍
n
-
1
/
2
2
j
(
1
/
2
-
1
/
p
+
d
)
2
-
j
d
(
r
/
p
)
2
-
j
(
s
+
1
/
2
-
1
/
r
)
(
r
/
p
)
×
(
n
ln
n
)
r
/
2
p
=
n
-
(
p
-
r
)
/
2
p
(
ln
n
)
-
r
/
2
p
∑
j
=
j
0
j
1
‍
2
-
j
ξ
.
Observing that e
21
≲
e
23, one obtains that
(86)
e
21
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
c
4
n
-
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.
(iii)
Finally, we estimate(87)
e
22
≔
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
×
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
1
/
p
.Let 1
<
q
′, q
<
∞ and 1
/
q
+
1
/
q
′
=
1. Thanks to Jensen’s inequality and Höler’s inequality, one has
(88)
e
22
=
E
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
1
/
p
≤

∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
∑
k
‍
E
(
|
β
^
j
k
-
β
j
k
|
p
I
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
)
1
/
p
≤
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
×
(
E
I
q
′
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
1
/
q
′
∑
k
‍
(
E
|
β
^
j
k
-
β
j
k
|
q
p
)
1
/
q
×
(
E
I
q
′
{
|
β
^
j
k
-
β
j
k
|
>
λ
2
}
)
1
/
q
′
)
1
/
p
≤
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
×
(
P
(
|
β
^
j
k
-
β
j
k
|
>
λ
2
)
)
1
/
q
′
∑
k
‍
(
E
|
β
^
j
k
-
β
j
k
|
q
p
)
1
/
q
×
(
P
(
|
β
^
j
k
-
β
j
k
|
>
λ
2
)
)
1
/
q
′
)
1
/
p
.
According to Lemmas 13 and 15, one knows that
(89)
e
22
≤
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
-
1
/
p
)
(
2
j
n
-
p
/
2
2
j
d
p
2
-
ω
j
/
q
′
)
1
/
p
=
n
-
1
/
2
∑
j
=
j
0
j
1
‍
2
j
(
1
/
2
+
d
-
ω
/
p
q
′
)
.
Choosing large enough ω such that 1
/
2
+
d
<
ω
/
p
q
′, one gets
(90)
e
22
≲
2
n
-
1
/
2
2
j
0
(
1
/
2
+
d
-
ω
/
p
q
′
)
≤
n
-
1
/
2
2
j
0
(
1
/
2
+
d
)
.
Taking 2
j
0 as in (70), one has
(91)
e
22
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
n
-
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
≤
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.
Putting (81), (86), and (91) together, one obtains that
(92)
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
c
4
n
-
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.Theorem 17.
Let the scaling functionϕ
(
x
) be orthonormal, compactly supported, and t-regular. If f
^
X
,
n
o
n
(
d
)
(
x
) is the nonlinear wavelet estimator defined in (57), and the assumptions (70), (71), and (81) are satisfied. Then for any 1
≤
r
<
p
<
∞, 1
/
r
<
s
<
t
-
d, one has
(93)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
X
,
non
(
d
)
-
f
X
(
d
)
∥
p
≲
{
(
ln
n
)
θ
1
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
θ
2
n
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
where s
′
=
s
-
1
/
r
+
1
/
p, θ
1, θ
2 are positive constants.Proof.
By the definition off
^
X
,
non
(
d
) in (57) and the expansion of f
X
(
d
) in (2), one has
(94)
f
^
X
,
non
(
d
)
-
f
X
(
d
)
=
∑
k
‍
(
α
^
j
0
k
-
α
j
0
k
)
ϕ
j
0
k
+
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
+
P
j
1
+
1
f
X
(
d
)
-
f
X
(
d
)
.
Hence,
(95)
E
∥
f
^
X
,
non
(
d
)
-
f
X
(
d
)
∥
p
≤
E
∥
∑
k
‍
(
α
^
j
0
k
-
α
j
0
k
)
ϕ
j
0
k
∥
p
+
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
+
∥
P
j
1
+
1
f
X
(
d
)
-
f
X
(
d
)
∥
p
=
:
e
1
+
e
2
+
e
3
,
where
(96)
e
1
:
=
E
∥
∑
k
‍
(
α
^
j
0
k
-
α
j
0
k
)
ϕ
j
0
k
∥
p
,
e
2
≔
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
,
e
3
:
=
∥
P
j
1
+
1
f
X
(
d
)
-
f
X
(
d
)
∥
p
.
Firstly, we estimate
(97)
e
1
:
=
E
∥
∑
k
‍
(
α
^
j
0
k
-
α
j
0
k
)
ϕ
j
0
k
∥
p
.
Thanks to Lemma 3 and Jensen’s inequality, one obtains that
(98)
e
1
≲
2
j
0
(
1
/
2
-
1
/
p
)
E
(
∑
k
‍
|
α
^
j
0
k
-
α
j
0
k
|
p
)
1
/
p
≤
2
j
0
(
1
/
2
-
1
/
p
)
(
∑
k
‍
E
|
α
^
j
0
k
-
α
j
0
k
|
p
)
1
/
p
.
Due to Lemma 6, one has E
|
α
^
j
0
k
-
α
j
0
k
|
p
≲
n
-
p
/
2
2
j
d
p. Since f
X
(
d
)
(
x
) and ϕ
(
x
) are compactly supported, then the number of elements in {
k
:
α
j
0
k
≠
0
} is O
(
2
j
0
). Therefore,
(99)
e
1
≲
2
j
0
(
1
/
2
-
1
/
p
)
(
2
j
0
n
-
p
/
2
2
j
0
d
p
)
1
/
p
=
n
-
1
/
2
2
j
0
(
1
/
2
+
d
)
.
From (70), one has
(100)
e
1
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
n
-
(
s
′
+
2
d
/
p
)
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
≤
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.
Next, we estimate term
(101)
e
3
:
=
∥
P
j
1
+
1
f
X
(
d
)
-
f
X
(
d
)
∥
p
.
When r
<
p, s
′
=
s
-
1
/
r
+
1
/
p, one has B
~
r
q
s
↪
B
~
p
q
s
′, thanks to the Besov spaces embedding theorem. By Lemma 4, one has
(102)
∥
P
j
1
+
1
f
X
(
d
)
-
f
X
(
d
)
∥
p
≲
2
-
j
1
s
′
.
Taking 2
j
1 as in (71), it can be proved that
(103)
e
3
≲
{
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
≤
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.
Finally, we estimate
(104)
e
2
:
=
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
.
Using Lemma 16, one has
(105)
E
∥
∑
j
=
j
0
j
1
‍
∑
k
‍
(
β
^
j
k
*
-
β
j
k
)
ψ
j
k
∥
p
≲
{
(
ln
n
)
c
3
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
c
4
n
-
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.
By (100), (103), and (105), one obtains that
(106)
sup
f
X
(
d
)
∈
B
~
r
q
s
(
A
,
L
)
E
∥
f
^
X
,
non
(
d
)
-
f
X
(
d
)
∥
p
≲
{
(
ln
n
)
θ
1
n
-
s
/
(
2
(
s
+
d
)
+
1
)
,
r
>
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
)
θ
2
n
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
=
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
,
(
ln
n
n
)
s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
,
r
<
(
1
+
2
d
)
p
2
(
s
+
d
)
+
1
.Remark 18.
Sincer
<
p implies that s
′
=
s
-
1
/
r
+
1
/
p, one can easily find s
′
/
(
2
(
s
-
1
/
r
+
d
)
+
1
)
>
s
′
/
(
2
(
s
′
+
d
)
+
1
). Then, Theorems 17 and 7 tell us that the nonlinear estimator does better than the linear one. Moreover, Theorems 17 and 11 show that our nonlinear one attains the optimal. covergence rateCorollary 19.
Whenr
<
p, the nonlinear estimator (57) attains the optimal covergence rate up to a ln
n factor.Remark 20.
Ifd
=
0, r
=
p
=
2, our estimation reduces to Ramírez and Vidakovic’s result [14].Remark 21.
Ifd
=
0, our results can be considered as the natural extension of paper [10].Remark 22.
Ifg
(
x
)
≡
1, model (1) reduces to the samples without any noise; our results reduce to the estimations in [13]. Even for p
/
(
2
(
s
+
d
)
+
1
)
<
r
<
p, our result is also optimal covergence rate.



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*Source: 512634-2014-01-30.xml*