Adaptive Wavelet Estimations in the Convolution Structure Density Model

Abstract

Using kernel methods, Lepski and Willer study a convolution structure density model and establish adaptive and optimal $L^p$ risk estimations over an anisotropic Nikol’skii space (Lepski, O.; Willer, T. Oracle inequalities and adaptive estimation in the convolution structure density model. Ann. Stat. 2019, 47, 233–287). Motivated by their work, we consider the same problem over Besov balls by wavelets in this paper and first provide a linear wavelet estimate. Subsequently, a non-linear wavelet estimator is introduced for adaptivity, which attains nearly-optimal convergence rates in some cases.

1. Introduction

The estimation of a probability density from independent and identically distributed (i.i.d.) random observations $X_1,X_2,\cdots ,X_n$ of X is a classical problem in statistics. The representative work is Donoho et al. [1], they established an adaptive and nearly-optimal estimate (up to a logarithmic factor) over Besov spaces using wavelets.

However, the observed data are always polluted by noises in many real-life applications. One of the important problems is the density estimation with an additive noise. Let $Z_1,Z_2,\cdots ,Z_n$ be i.i.d. random variables and have the same distribution as

$$\begin{array}{c}\hfill Z=X+Y,\end{array}$$

(1)

where X denotes a real-valued random variable with unknown probability density function f and Y stands for an independent random noise (error) with a known probability density g. The problem is to estimate f by $Z_1,Z_2,\cdots ,Z_n$ in some sense. Moreover, it is also called a denconvolution problem (model), because the density h of Z equals the convolution of f and g. Fan and Koo [2] studied the MISE performance ($L^2$-risk) of linear wavelet deconvolution estimator over a Besov ball. The $L^{\infty }$ risk optimal wavelet estimations were investigated by Lounici and Nickl [3]. Furthermore, Li and Liu [4] provided $L^p(1\le p\le \infty )$ risk optimal deconvolution estimations using wavelet bases.

In this paper, we consider a generalized deconvolution model introduced by Lepski & Willer [5,6]. More precisely, let $(\mathsf{\Omega },\mathcal{F},P)$ be a probability space and $Z_1,Z_2,\cdots ,Z_n$ be i.i.d. random variables having the same distribution as

$$\begin{array}{c}\hfill Z=X+\epsilon Y,\end{array}$$

(2)

where the symbols X and Y are same as model (1), f and g are the corresponding densities respectively. Moreover, the biggest difference with model (1) is that a Bernoulli random variable $\epsilon \in \{0,1\}$ with $P\{\epsilon =1\}=\alpha$ is added in (2), and $\alpha \in [0,1]$ is known. The problem is also to estimate f by the observed data $Z_1,Z_2,\cdots ,Z_n$ in some sense.

When $\alpha =1$, model (2) reduces to the deconvolution one (see [2,3,4,7,8] et al.), while $\alpha =0$ corresponds to the classical density model with no errors (see [1,9,10,11] et al.). Clearly, the density function h of Z in (2) satisfies

$$\begin{array}{c}\hfill h=(1-\alpha )f+\alpha f\ast g.\end{array}$$

Here, $f\ast g$ stands for the convolution of f and g. Furthermore, when the function $G_{\alpha }\left(t\right):=1-\alpha +\alpha g^{ft}\left(t\right)\ne 0$ for $t\in \mathbb{R}$, we have

$$f^{ft}\left(t\right)={[(1-\alpha )+\alpha g^{ft}\left(t\right)]}^{-1}{h}^{ft}\left(t\right)={\left(G_{\alpha }\left(t\right)\right)}^{-1}{h}^{ft}\left(t\right),$$

where g, $\alpha$ are known and $f^{ft}$ is the Fourier transform of $f\in L^1\left(\mathbb{R}\right)$ given by

$$f^{ft}\left(t\right):={\int }_{\mathbb{R}}f\left(x\right)e^{-itx}dx.$$

Based on the model (2) with some mild assumptions on $G_{\alpha }$, Lepski and Willer [5] provided a lower bound estimation over $L^p$ risk on an anistropic Nikol’skii space. Moreover, they investigated an adaptive and optimal $L^p$ estimate by using kernel method in Ref. [6]. Recently, Wu et al. [12] established a pointwise lower bound estimation for model (2) under the local Hölder condition.

When compared with the classical kernel estimation of density functions, the wavelet estimations provide more local information and fast algorithm [13]. We will consider the $L^p(1\le p<\infty )$ risk estimations under the model (2) over Besov balls by using wavelets and expect to obtain the corresponding convergence rates.

The same as Assumption 4 in [6], we also need the following condition on Y,

$$\begin{array}{c}\hfill |G_{\alpha }{\left(t\right)|\gtrsim (1+|t|}^2{)}^{-\frac{\beta \left(\alpha \right)}{2}}\end{array}$$

(3)

with $\beta \left(\alpha \right)=\beta \ge 0$ for $\alpha =1$ and $\beta \left(\alpha \right)=0$ for others. It is reasonable, because it holds automatically for $\alpha =0$, while the same condition for $\alpha =1$ is necessary for the deconvolution estimations [4,7]. In addition, when $0<\alpha <\frac{1}{2}$, $\beta \left(\alpha \right)=0$ and $|G_{\alpha }\left(t\right)|\ge 1-\alpha -\alpha |g^{ft}\left(t\right)|\gtrsim 1$ thanks to $\Vert g^{ft}{\Vert }_{\infty }\le 1$. In fact, the condition (3) is necessary to prove Lemmas 2 and 3 in Section 2. Here and after, $A\lesssim B$ denotes $A\le cB$ for a fixed constant $c>0$; $A\gtrsim B$ means $B\lesssim A$; $A\sim B$ stands for both $A\lesssim B$ and $A\gtrsim B$.

It is well-known that the wavelet estimation depends on an orthonormal wavelet expansion in $L^2\left(\mathbb{R}\right)$, even in $L^p\left(\mathbb{R}\right)$. Let $\{V_j:j\in \mathbb{N}\}$ be a classical Multiresolution Analysis of $L^2\left(\mathbb{R}\right)$ with scaling function $\phi$ and $\psi$ being the corresponding wavelet. Subsequently, for $f\in L^2\left(\mathbb{R}\right),$

$$\begin{array}{c}\hfill f=\sum _{k\in \mathbb{Z}}{\alpha }_{j_{0}k}{\phi }_{j_{0}k}+\sum _{j=j_0}^{\infty }\sum _{k\in \mathbb{Z}}{\beta }_{jk}{\psi }_{jk},\end{array}$$

(4)

where ${\alpha }_{jk}:=⟨f,{\phi }_{jk}⟩,{\beta }_{jk}:=⟨f,{\psi }_{jk}⟩$ and ${\vartheta }_{jk}(·):=2^{\frac{j}{2}}\vartheta (2^j·-k)(\vartheta =\phi or\psi )$. A scaling function $\phi$ is called m-regular $(m\in \mathbb{N})$, if $\phi \in {\mathbf{C}}^m\left(\mathbb{R}\right)$ and $|{\phi }^{\left(k\right)}{\left(x\right)|\lesssim (1+|x|}^2{)}^{-l}$ for each $l\in \mathbb{N}$ ($k=0,1,\cdots ,m$). Clearly, the m-regularity of $\phi$ implies that of the corresponding $\psi$, and $|{\phi }^{ft}{\left(t\right)|\lesssim (1+|t|}^2{)}^{-\frac{m}{2}}$ due to the integration by parts. An important example is Daubechies’ function $D_{2N}$ with N large enough.

As usual, let $P_j$ be the orthogonal projection from $L^2\left(\mathbb{R}\right)$ onto $V_j$,

$$\begin{array}{c}\hfill P_{j}f:=\sum _{k\in \mathbb{Z}}{\alpha }_{jk}{\phi }_{jk}.\end{array}$$

(5)

If $\phi$ is m-regular, then $P_{j}f$ is well-defined for $f\in L^p\left(\mathbb{R}\right)$. Moreover, the identity (4) holds in $L^p\left(\mathbb{R}\right)$ for $1\le p\le \infty$.

The following lemma is needed for later discussions.

Lemma 1

([13]). Let ϑ be an orthogonal scaling function or a wavelet satisfying m-regularity. Subsequently, there exist $C_2\ge C_1>0$, such that, for $\lambda =\{{\lambda }_k\}\in l^p\left(\mathbb{Z}\right)$ and $1\le p\le \infty$,

$$C_1{2}^{j(\frac{1}{2}-\frac{1}{p})}{\Vert \lambda \Vert }_{l_p}\le \Vert \sum _{k\in \mathbb{Z}}{\lambda }_k{\vartheta }_{jk}{\Vert }_p\le C_2{2}^{j(\frac{1}{2}-\frac{1}{p})}{\Vert \lambda \Vert }_{l_p}.$$

One of the advantages of wavelet bases is that they can characterize Besov spaces, which contain the $L^2$-Sobolev spaces and Hölder spaces as special examples.

Proposition 1

([13]). Let scaling function φ be m-regular with $m>s>0$ and ψ be the corresponding wavelet. Afterwards, for $r,q\in [1,\infty ]$ and $f\in L^r\left(\mathbb{R}\right)$, the following conditions are equivalent:

(i). 

$f\in B_{r,q}^s\left(\mathbb{R}\right)$;

(ii). 

$\{2^{js}\Vert P_{j}f-f{{\Vert }_r\}}_{j\ge 0}\in l_q$; and,

(iii). 

$\Vert {\alpha }_{0·}{\Vert }_{l_r}+\Vert (2^{j(s+\frac{1}{2}-\frac{1}{r})}\Vert {\beta }_{j·}{\Vert }_{l_r}{)\Vert }_{l_q}<\infty .$

The Besov norm can be defined by

$${\Vert f\Vert }_{B_{r,q}^s}:=\Vert {\alpha }_{j_0·}{\Vert }_{l_r}+\Vert (2^{j(s+\frac{1}{2}-\frac{1}{r})}\Vert {\beta }_{j·}{\Vert }_{l_r}{)\Vert }_{l_q}.$$

When $s>0$ and $1\le r,p,q\le \infty$, it is well-known that

(1)

$B_{r,q}^s\hookrightarrow B_{r,\infty }^s\hookrightarrow B_{\infty ,\infty }^{s-\frac{1}{r}}$ for $s>\frac{1}{r}$;

(2)

$B_{r,q}^s\hookrightarrow B_{p,q}^{s^{\prime }}$ for $r\le p$ and $s-\frac{1}{r}=s^{\prime }-\frac{1}{p}$,

where $A\hookrightarrow B$ stands for a Banach space A continuously embedded in another Banach space B. More precisely, ${\Vert u\Vert }_B\le c{\Vert u\Vert }_A(u\in A)$ holds for some $c>0$.

In this paper, we use the notation $B_{r,q}^s(L,M)$ with some constants $L,M>0$ to stand for a Besov ball, i.e.,

$$B_{r,q}^s(L,M):=\{f\in L^r{\left(\mathbb{R}\right),f\mathrm{is}\mathrm{a}\mathrm{density},\Vert f\Vert }_{B_{r,q}^s}\le L\mathrm{and}\mathrm{supp}f\subseteq [-M,M]\}.$$

Next, we will estimate f with $L^p$ risk by constructing wavelet estimators from the observed data $Z_1,Z_2,\cdots ,Z_n$. To introduce wavelet estimators, we take $\phi$ having compact support and m-regularity with $m\ge \beta \left(\alpha \right)+2$ in this paper. Moreover, denote

$$\begin{array}{c}\hfill {\widehat{\alpha }}_{jk}:=\frac{1}{n}\sum _{l=1}^n\left(K{\phi }_{jk}\right)\left(Z_l\right),\end{array}$$

where

$$\begin{array}{c}\hfill \left(K{\phi }_{jk}\right)\left(x\right):=\frac{1}{2\pi }{\int }_{\mathbb{R}}{e}^{-itx}\overline{{\left({\phi }_{jk}\right)}^{ft}\left(t\right)}{G}_{\alpha }^{-1}\left(t\right)dt\end{array}$$

(6)

and $K{\psi }_{jk}$ is defined by the way. Clearly, $E{\widehat{\alpha }}_{jk}={\alpha }_{jk}$ due to the Plancherel formula. Subsequently, the linear wavelet estimator is given by

$$\begin{array}{c}\hfill {\widehat{f}}_n^{lin}:=\sum _{k\in {\Lambda }_{j_0}}{\widehat{\alpha }}_{j_{0}k}{\phi }_{j_{0}k},\end{array}$$

(7)

where ${\Lambda }_j:=\{k\in \mathbb{Z},\mathrm{supp}f\cap \mathrm{supp}{\phi }_{jk}\ne \varnothing \}\cup \{k\in \mathbb{Z},\mathrm{supp}f\cap \mathrm{supp}{\psi }_{jk}\ne \varnothing \}$. In particular, the cardinality of ${\Lambda }_j$ satisfies that $|{\Lambda }_j|\sim 2^j$, when f and $\phi$ have compact supports.

Now, we are in a position to state the first result of this paper.

Theorem 1.

For $r,p\in [1,+\infty ),q\in [1,+\infty ]$ and $\frac{1}{r}<s<m$, the estimator ${\widehat{f}}_n^{lin}$ in (7) with $2^{j_0}\sim n^{\frac{1}{2s^{\prime }+2\beta \left(\alpha \right)+1}}$ satisfies

$$\begin{array}{c}\hfill \underset{f\in B_{r,q}^s(L,M)}{sup}E{\Vert {\widehat{f}}_n^{lin}-f\Vert }_p^p\lesssim n^{-\frac{s^{\prime }p}{2s^{\prime }+2\beta \left(\alpha \right)+1}},\end{array}$$

where $s^{\prime }=s-{(\frac{1}{r}-\frac{1}{p})}_{+}$ and $a_{+}=max\{a,0\}$.

Remark 1.

When $\alpha =1$, $\beta \left(\alpha \right)=\beta$, the conclusion of Theorem 1 reduces to Theorem 3 of Li & Liu [4].

Note that the estimator ${\widehat{f}}_n^{lin}$ is non-adaptive, because the choice of $j_0$ depends on the unknown parameter s. To obtain an adaptive estimate, define

$$\begin{array}{c}\hfill {\widehat{\beta }}_{jk}:=\frac{1}{n}\sum _{l=1}^n\left(K{\psi }_{jk}\right)\left(Z_l\right)\mathrm{and}{\tilde{\beta }}_{jk}:={\widehat{\beta }}_{jk}{I}_{\{|{\widehat{\beta }}_{jk}|>{\tau }_{j,n}\}}.\end{array}$$

(8)

Here, ${\tau }_{j,n}=c\gamma 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$ and the constants $c,\gamma$ will be determined later on. Subsequently, the non-linear wavelet estimator is defined by

$$\begin{array}{c}\hfill {\widehat{f}}_n^{non}:=\sum _{k\in {\Lambda }_{j_0}}{\widehat{\alpha }}_{j_{0}k}{\phi }_{j_{0}k}+\sum _{j=j_0}^{j_1}\sum _{k\in {\Lambda }_j}{\tilde{\beta }}_{jk}{\psi }_{jk},\end{array}$$

(9)

where $j_0$, $j_1$ are positive integers satisfying $2^{j_0}\sim n^{\frac{1}{2m+2\beta \left(\alpha \right)+1}}$ and $2^{j_1}\sim \frac{n}{lnn}$ respectively. Clearly, $j_0$ and $j_1$ do not depend on the unknown parameters $s,r,q$, which means that the estimator ${\widehat{f}}_n^{non}$ in (9) is adaptive.

Theorem 2.

Let $r,p\in [1,+\infty ),q\in [1,+\infty ]$ and $\frac{1}{r}<s<m$. Then the estimator ${\widehat{f}}_n^{non}$ in (9) satisfies

$$\begin{array}{c}\hfill \underset{f\in B_{r,q}^s(L,M)}{sup}E{\Vert {\widehat{f}}_n^{non}-f\Vert }_p^p\lesssim {(lnn)}^p{\left(\frac{lnn}{n}\right)}^{\theta p},\end{array}$$

where $\theta :=min\{\frac{s}{2s+2\beta \left(\alpha \right)+1},\frac{s-\frac{1}{r}+\frac{1}{p}}{2(s-\frac{1}{r})+2\beta \left(\alpha \right)+1}\}.$

Remark 2.

When $\alpha =0$, $\beta \left(\alpha \right)=0$ and $\theta =min\{\frac{s}{2s+1},\frac{s-\frac{1}{r}+\frac{1}{p}}{2(s-\frac{1}{r})+1}\}$, the convergence rate of Theorem 2 coincides with that of Theorem 3 in Donoho et al. [1]. On the other hand, $\beta \left(\alpha \right)=\beta$ and $\theta =min\{\frac{s}{2s+2\beta +1},\frac{s-\frac{1}{r}+\frac{1}{p}}{2(s-\frac{1}{r})+2\beta +1}\}$ for the case $\alpha =1$, while the conclusion of Theorem 4 in Li & Liu [4] can follow directly from this theorem.

Remark 3.

When comparing the result of Theorem 2 with Theorem 1, we find easily that for the case $r\le p$, the convergence rate of non-linear estimator is better than that of the linear one with $n^{-\frac{s^{\prime }p}{2s^{\prime }+2\beta \left(\alpha \right)+1}}$ and $s^{\prime }=s-\frac{1}{r}+\frac{1}{p}$.

Remark 4.

The convergence rates of Theorem 2 with the cases $\alpha =0$ and $\alpha =1$ are nearly-optimal (up to a logarithmic factor) by Donoho et al. [1] and Li & Liu [4] respectively. However, it is not clear whether the estimation in Theorem 2 is optimal (nearly-optimal) or not for $\alpha \in (0,1)$. Therefore, one of our future work is to determine a low bound estimate for model (2) with $\alpha \in (0,1)$. This problem may be much more complicated than the cases of $\alpha =0$ and $\alpha =1$.

2. Preliminaries

This section is devoted to introduce some useful lemmas. The following inequality is necessary in the proof of Lemma 2.

Rosenthal’s inequality ([13]). Let $p>0$ and $X_1,X_2,\cdots ,X_n$ be the independent random variables such that $E{X}_l=0$ and $E|X_l{|}^p<\infty$$(l=1,2,\cdots ,n)$. Subsequently, there exists $C\left(p\right)>0$, such that

$$E{\left|\sum _{l=1}^n{X}_l\right|}^p\le C\left(p\right)\left\{\sum _{l=1}^{n}E{|X_l|}^p{I}_{\{p>2\}}+{\left(\sum _{l=1}^{n}E{X}_l^2\right)}^{\frac{p}{2}}\right\}.$$

Lemma 2.

Let $1\le 2^j\le n$ and ${\Vert f\Vert }_{\infty }\lesssim 1$. Then for $p\in [1,+\infty )$,

$$\begin{array}{c}\hfill E|{\widehat{\alpha }}_{jk}-{\alpha }_{jk}{|}^p\lesssim n^{-\frac{p}{2}}{2}^{j\beta \left(\alpha \right)p}andE{|{\widehat{\beta }}_{jk}-{\beta }_{jk}|}^p\lesssim n^{-\frac{p}{2}}{2}^{j\beta \left(\alpha \right)p}.\end{array}$$

Proof. 

Obviously, one only needs to prove the first inequality and the second one is similar. Define ${\xi }_l:=\left(K{\phi }_{jk}\right)\left(Z_l\right)-E\left(K{\phi }_{jk}\right)\left(Z_l\right)$ ($l=1,\cdots ,n$). Subsequently, ${\{{\xi }_l\}}_{l=1}^n$ are i.i.d. samples and $E{\xi }_l=0$ ($l=1,\cdots ,n$). By the definitions of ${\alpha }_{jk}$ and ${\widehat{\alpha }}_{jk}$, $E{\widehat{\alpha }}_{jk}={\alpha }_{jk}$ and

$$\begin{array}{c}\hfill E|{\widehat{\alpha }}_{jk}-{\alpha }_{jk}{|}^p=\frac{1}{n^p}E{\left|\sum _{l=1}^n[\left(K{\phi }_{jk}\right)\left(Z_l\right)-E\left(K{\phi }_{jk}\right)\left(Z_l\right)]\right|}^p=\frac{1}{n^p}E{\left|\sum _{l=1}^n{\xi }_l\right|}^p.\end{array}$$

(10)

According to (6), one obtains that

$$|\left(K{\phi }_{jk}\right)\left(Z_l\right)|\lesssim {\int }_{\mathbb{R}}|{\left({\phi }_{jk}\right)}^{ft}\left(t\right)G_{\alpha }^{-1}\left(t\right)|dt=2^{-\frac{j}{2}}{\int }_{\mathbb{R}}|{\phi }^{ft}\left(2^{-j}t\right)G_{\alpha }^{-1}\left(t\right)|dt.$$

This with (3), $m(m\ge \beta (\alpha )+2)$ regularity of $\phi$ and $2^j\ge 1$ shows

$$\begin{array}{c}\hfill |(K{\phi }_{jk})(Z_l)|\lesssim 2^{\frac{j}{2}}{\int }_{\mathbb{R}}{(1+|t|}^2{)}^{-\frac{m}{2}}(1+|2^{j}t{{|}^2)}^{\frac{\beta \left(\alpha \right)}{2}}dt\lesssim 2^{j(\frac{1}{2}+\beta \left(\alpha \right))}.\end{array}$$

(11)

Hence, for $l=1,2,\cdots ,n,$

$$\begin{array}{c}\hfill \Vert {\xi }_l{\Vert }_{\infty }\lesssim 2^{j(\frac{1}{2}+\beta \left(\alpha \right))}.\end{array}$$

(12)

On the other hand, ${\Vert h\Vert }_{\infty }={\Vert (1-\alpha )f+\alpha (f\ast g)\Vert }_{\infty }\lesssim 1$ follows from ${\Vert f\Vert }_{\infty }\lesssim 1$. Afterwards,

$$E|{\xi }_l{|}^2\le E{(K{\phi }_{jk})}^2(Z_l)={\int }_{\mathbb{R}}|K{\phi }_{jk}\left(z\right){|}^{2}h\left(z\right)dz\lesssim {\int }_{\mathbb{R}}{|K{\phi }_{jk}\left(z\right)|}^{2}dz.$$

Furthermore, $E|{\xi }_l{|}^2\lesssim {\int }_{\mathbb{R}}{|{\left({\phi }_{jk}\right)}^{ft}\left(z\right)G_{\alpha }^{-1}\left(z\right)|}^{2}dz$ due to the Plancherel formula. The same arguments as (11) imply that

$$\begin{array}{c}\hfill E|{\xi }_l{|}^2\lesssim {\int }_{\mathbb{R}}{(1+|t|}^2{)}^{-m}(1+|2^{j}t{{|}^2)}^{\beta \left(\alpha \right)}dt\lesssim 2^{2j\beta \left(\alpha \right)}.\end{array}$$

(13)

By Rosenthal’s inequality, $E|{\sum }_{l=1}^n{\xi }_l{|}^p\lesssim {\sum }_{l=1}^{n}E{|{\xi }_l|}^p{I}_{\{p>2\}}+{\left({\sum }_{l=1}^{n}E{\xi }_l^2\right)}^{\frac{p}{2}}.$ This with (12) and (13) and $2^j\le n$ shows

$$\begin{array}{c}\hfill E{\left|\sum _{l=1}^n{\xi }_l\right|}^p\lesssim n^{\frac{p}{2}}{2}^{j\beta \left(\alpha \right)p}[{\left(n^{-1}{2}^j\right)}^{\frac{p}{2}-1}{I}_{\{p>2\}}+1]\lesssim n^{\frac{p}{2}}{2}^{j\beta \left(\alpha \right)p}.\end{array}$$

(14)

Finally, the desired conclusion is concluded from (10) and (14). □

We state another classical inequality, before giving the proof of Lemma 3.

Bernstein’s inequality ([13]).Let $X_1,X_2,\cdots ,X_n$ be independent random variables with $E{X}_l=0,E{X}_l^2\le M$ and $|X_l{|\le \Vert X\Vert }_{\infty }$$(l=1,2,\cdots ,n)$. Then for each $ϵ>0$,

$$P\left\{\left|\frac{1}{n}\sum _{l=1}^n{X}_l\right|>ϵ\right\}\le 2exp\left\{-\frac{n{ϵ}^2}{2\left(M+{\Vert X\Vert }_{\infty }ϵ/3\right)}\right\}.$$

Lemma 3.

If $j{2}^j\le n,2^j\ge 1$ and ${\Vert f\Vert }_{\infty }\lesssim 1$, then there exists some constat $c>0$ such that for any $\gamma >0$,

$$P\left\{|{\widehat{\beta }}_{jk}-{\beta }_{jk}|>\frac{{\tau }_{j,n}}{2}\right\}\lesssim 2^{-\gamma j},$$

where ${\tau }_{j,n}=c\gamma 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$.

Proof. 

Denote ${\eta }_l:=\left(K{\psi }_{jk}\right)\left(Z_l\right)-E\left(K{\psi }_{jk}\right)\left(Z_l\right),l=1,\cdots ,n$. Then ${\{{\eta }_l\}}_{l=1}^n$ are i.i.d., $E{\eta }_l=0$ and $|{\widehat{\beta }}_{jk}-{\beta }_{jk}|=|\frac{1}{n}{\sum }_{l=1}^n{\eta }_l|$ by (8). Moreover, the same arguments as (12) and (13) show

$$\Vert {\eta }_l{\Vert }_{\infty }\le H_1{2}^{j(\frac{1}{2}+\beta \left(\alpha \right))}\mathrm{and}E{\eta }_l^2\le H_2{2}^{2j\beta \left(\alpha \right)}$$

with some positive constants $H_1,H_2$.

According to Bernstein’s inequality,

$$\begin{array}{c}\hfill P\left\{|{\widehat{\beta }}_{jk}-{\beta }_{jk}|>\frac{{\tau }_{j,n}}{2}\right\}\le 2exp\left\{-\frac{n{\tau }_{j,n}^2}{8[H_2{2}^{2j\beta \left(\alpha \right)}+H_1{2}^{j(\frac{1}{2}+\beta \left(\alpha \right))}{\tau }_{j,n}/6]}\right\}.\end{array}$$

(15)

Note that for $j{2}^j\le n$,

$$8[H_2{2}^{2j\beta (\alpha )}+H_1{2}^{j(\frac{1}{2}+\beta (\alpha ))}{\tau }_{j,n}/6]\le H{2}^{2j\beta (\alpha )}(1+c\gamma \sqrt{\frac{j{2}^j}{n}})\le (1+c\gamma )H{2}^{2j\beta (\alpha )}$$

with some positive constant H. Hence,

$$\frac{n{\tau }_{j,n}^2}{8[H_2{2}^{2j\beta \left(\alpha \right)}+H_1{2}^{j(\frac{1}{2}+\beta \left(\alpha \right))}{\tau }_{j,n}/6]}\ge \frac{n{c}^2{\gamma }^{2}j{n}^{-1}{2}^{2j\beta \left(\alpha \right)}}{(1+c\gamma )H{2}^{2j\beta \left(\alpha \right)}}=\frac{c^2{\gamma }^{2}j}{(1+c\gamma )H}.$$

Furthermore, (15) reduces to

$$P\left\{|{\widehat{\beta }}_{jk}-{\beta }_{jk}|>\frac{{\tau }_{j,n}}{2}\right\}\le 2e^{-\frac{c^2{\gamma }^{2}j}{(1+c\gamma )H}}\lesssim 2^{-\gamma j}$$

by choosing $c\ge max\{2Hln2,\sqrt{2H{\gamma }^{-1}ln2}\}$. This completes the proof. □

3. Proofs of Main Results

We shall show the proofs of Theorems 1 and 2 in this section.

Proof of Theorem 1.

It is sufficient to prove the case for $r\le p$. In fact, when $r>p$ and f has a compact support, ${\widehat{f}}_n^{lin}$ does because of $\phi$ having the same property. Subsequently, it follows from Hölder inequality and Jensen’s inequality that

$$\begin{array}{c}\hfill \underset{f\in B_{r,q}^s(L,M)}{sup}E\Vert {\widehat{f}}_n^{lin}{-f\Vert }_p^p\lesssim \underset{f\in B_{r,q}^s(L,M)}{sup}(E\Vert {\widehat{f}}_n^{lin}-f{{\Vert }_r^r)}^{\frac{p}{r}}\lesssim n^{-\frac{s^{\prime }p}{2s^{\prime }+2\beta \left(\alpha \right)+1}}.\end{array}$$

According to (5) and (7), one easily finds that

$$\begin{array}{c}\hfill E\Vert {\widehat{f}}_n^{lin}{-f\Vert }_p^p\lesssim \Vert P_{j_0}{f-f\Vert }_p^p+E{\Vert {\widehat{f}}_n^{lin}-E{\widehat{f}}_n^{lin}\Vert }_p^p.\end{array}$$

(16)

Clearly, by Proposition 1, $\Vert P_{j_0}{f-f\Vert }_p^p\lesssim 2^{-j_0{s}^{\prime }p}$ thanks to the well-known embedding theorem $B_{r,q}^s\hookrightarrow B_{p,q}^{s^{\prime }}$ for $r\le p$.

On the other hand, $s>\frac{1}{r}$ implies ${\Vert f\Vert }_{\infty }\lesssim 1$ and Lemma 2 tells that $E|{\widehat{\alpha }}_{jk}-{\alpha }_{jk}{|}^p\lesssim 2^{j\beta \left(\alpha \right)p}{n}^{-\frac{p}{2}}$. This with Lemma 1 and $|{\Lambda }_{j_0}|\sim 2^{j_0}$ shows that

$$\begin{array}{c}\hfill E\Vert {\widehat{f}}_n^{lin}-E{\widehat{f}}_n^{lin}{\Vert }_p^p\lesssim 2^{j_0(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_{j_0}}E{|{\widehat{\alpha }}_{j_{0}k}-{\alpha }_{j_{0}k}|}^p\lesssim 2^{j_0(\beta \left(\alpha \right)+\frac{1}{2})p}{n}^{-\frac{p}{2}}.\end{array}$$

(17)

Finally, (16) reduces to

$$E\Vert {\widehat{f}}_n^{lin}{-f\Vert }_p^p\lesssim 2^{-j_0{s}^{\prime }p}+2^{j_0(\beta \left(\alpha \right)+\frac{1}{2})p}{n}^{-\frac{p}{2}}\lesssim n^{-\frac{s^{\prime }p}{2s^{\prime }+2\beta \left(\alpha \right)+1}}$$

because of $2^{j_0}\sim n^{\frac{1}{2s^{\prime }+2\beta \left(\alpha \right)+1}}$. The proof is done. □

Now, we give a proof of Theorem 2, which is the most important result.

Proof of Theorem 2.

The same arguments as the proof of Theorem 1, one only needs to prove the case for $r\le p$.

According to (4), (5) and (9), $E\Vert {\widehat{f}}_n^{non}{-f\Vert }_p^p\lesssim A_n+B_n+C_n$, where

$$A_n=E\Vert \sum _{k\in {\Lambda }_{j_0}}({\widehat{\alpha }}_{j_{0}k}-{\alpha }_{j_{0}k}){\phi }_{j_{0}k}{\Vert }_p^p,B_n=E{\Vert \sum _{j=j_0}^{j_1}\sum _{k\in {\Lambda }_j}({\tilde{\beta }}_{jk}-{\beta }_{jk}){\psi }_{jk}\Vert }_p^p$$

and $C_n={\Vert f-P_{j_1}f\Vert }_p^p.$

It is known that $B_{r,q}^s\hookrightarrow B_{p,q}^{s^{\prime }}$ for $r\le p$ with $s^{\prime }=s-\frac{1}{r}+\frac{1}{p}$. Hence, $C_n={\Vert f-P_{j_1}f\Vert }_p^p\lesssim 2^{-j_1{s}^{\prime }p}$ due to Proposition 1. Moreover, it follows from the choice of $2^{j_1}\sim \frac{n}{lnn}$, $s>\frac{1}{r}$ and $\theta :=min\{\frac{s}{2s+2\beta \left(\alpha \right)+1},\frac{s-\frac{1}{r}+\frac{1}{p}}{2(s-\frac{1}{r})+2\beta \left(\alpha \right)+1}\}$ that

$$C_n\lesssim {\left(\frac{lnn}{n}\right)}^{s^{\prime }p}\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}.$$

Similar to (17), one obtains

$$\begin{array}{c}\hfill A_n=E{\Vert \sum _{k\in {\Lambda }_{j_0}}({\widehat{\alpha }}_{j_{0}k}-{\alpha }_{j_{0}k}){\phi }_{j_{0}k}\Vert }_p^p\lesssim 2^{j_0(\beta \left(\alpha \right)+\frac{1}{2})p}{n}^{-\frac{p}{2}}\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}\end{array}$$

by $2^{j_0}\sim n^{\frac{1}{2m+2\beta \left(\alpha \right)+1}}$, $s<m$ and the definition of $\theta$.

Next, the main work of this proof is to estimate $B_n$. By Lemma 1,

$$B_n=E\Vert \sum _{j=j_0}^{j_1}\sum _{k\in {\Lambda }_j}({\tilde{\beta }}_{jk}-{\beta }_{jk}){\psi }_{jk}{\Vert }_p^p\lesssim j_1^{p-1}\sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}E{|{\tilde{\beta }}_{jk}-{\beta }_{jk}|}^p.$$

Define ${\widehat{B}}_j:=\{k:|{\widehat{\beta }}_{jk}|>{\tau }_{j,n}\},B_j:=\{k:|{\beta }_{jk}|>\frac{{\tau }_{j,n}}{2}\},C_j:=\{k:|{\beta }_{jk}|>2{\tau }_{j,n}\}$ and ${\xi }_{jk}:={\widehat{\beta }}_{jk}-{\beta }_{jk}$. Subsequently, $B_n\lesssim {(lnn)}^{p-1}{\sum }_{l=1}^{4}E{e}_l,$ where

$$\begin{array}{ccc}\hfill e_1& :=& \sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\xi }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j\cap B_j^c\}},\hfill \\ \hfill e_2& :=& \sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\xi }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j\cap B_j\}},\hfill \\ \hfill e_3& :=& \sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\beta }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j^c\cap C_j\}},\hfill \\ \hfill e_4& :=& \sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\beta }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j^c\cap C_j^c\}}\hfill \end{array}$$

with $A^c$ denoting the complement of A in $\mathbb{Z}$. Hence, it suffices to prove $E{e}_l\lesssim lnn{\left(\frac{lnn}{n}\right)}^{\theta p}(l=1,2,3,4)$.

To estimate $E{e}_1$, note that $|{\widehat{\beta }}_{jk}-{\beta }_{jk}|\ge \frac{{\tau }_{j,n}}{2}$ follows from $k\in {\widehat{B}}_j\cap B_j^c$. Moreover, $E|{\xi }_{jk}{|}^p{I}_{\{k\in {\widehat{B}}_j\cap B_j^c\}}\le (E|{\xi }_{jk}{|}^{2p}{)}^{\frac{1}{2}}{P}^{\frac{1}{2}}(|{\xi }_{jk}|>\frac{{\tau }_{j,n}}{2})$ by the Hölder inequality. Afterwards, for large $\gamma$,

$$\begin{array}{c}\hfill E{e}_1\lesssim \sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{n}^{-\frac{p}{2}}{2}^{j\beta \left(\alpha \right)p}{2}^{-\frac{\gamma j}{2}}\lesssim 2^{j_0(\frac{p}{2}+\beta \left(\alpha \right)p-\frac{\gamma }{2})}{n}^{-\frac{p}{2}}\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}\end{array}$$

thanks to $|{\Lambda }_j|\sim 2^j$, Lemmas 2 and 3.

For $e_3$, $|{\widehat{\beta }}_{jk}-{\beta }_{jk}|\ge \frac{{\tau }_{j,n}}{2}$ due to $k\in {\widehat{B}}_j^c\cap C_j$. Combining this with Lemma 3, one knows $E{I}_{\{k\in {\widehat{B}}_j^c\cap C_j\}}\le P(|{\xi }_{jk}|\ge \frac{{\tau }_{j,n}}{2})\lesssim 2^{-\gamma j}$. Therefore, by $|{\beta }_{jk}|\lesssim 2^{\frac{j}{2}}$, $|{\Lambda }_j|\sim 2^j$ and large $\gamma$,

$$\begin{array}{c}\hfill E{e}_3\lesssim \sum _{j=j_0}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{2}^{\frac{pj}{2}}{2}^{-\gamma j}\lesssim 2^{-(\gamma -p)j_0}\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}.\end{array}$$

In order to estimate $e_2$ and $e_4$, one defines $j_0^{\ast },j_1^{\ast }\in \mathbb{Z}$ satisfying

$$2^{j_0^{\ast }}\sim {\left(\frac{n}{lnn}\right)}^{\frac{1-2\theta }{2\beta \left(\alpha \right)+1}}\mathrm{and}{2}^{j_1^{\ast }}\sim {\left(\frac{n}{lnn}\right)}^{\frac{\theta }{s-\frac{1}{r}+\frac{1}{p}}}.$$

Recall that $\theta :=min\{\frac{s}{2s+2\beta \left(\alpha \right)+1},\frac{s-\frac{1}{r}+\frac{1}{p}}{2(s-\frac{1}{r})+2\beta \left(\alpha \right)+1}\}$, $2^{j_0}\sim n^{\frac{1}{2m+2\beta \left(\alpha \right)+1}}$ and $2^{j_1}\sim \frac{n}{lnn}$. Then $j_0\le j_0^{\ast }\le j_1^{\ast }\le j_1$ due to $\frac{1}{r}<s<m$ and $r\le p$.

To estimate $E{e}_2$, one divides $e_2$ into

$$e_2=(\sum _{j=j_0}^{j_0^{\ast }}+\sum _{j=j_0^{\ast }+1}^{j_1})2^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\xi }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j\cap B_j\}}:=e_{21}+e_{22}.$$

Similar to (17), according to $2^{j_0^{\ast }}\sim {\left(\frac{n}{lnn}\right)}^{\frac{1-2\theta }{2\beta \left(\alpha \right)+1}}$,

$$E{e}_{21}\lesssim 2^{j_0^{\ast }(\beta \left(\alpha \right)+\frac{1}{2})p}{n}^{-\frac{p}{2}}\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}.$$

Define

$$\omega :=\left(s+\beta \left(\alpha \right)+\frac{1}{2}\right)r-\left(\beta \left(\alpha \right)+\frac{1}{2}\right)p.$$

When $\omega \ge 0$, $\theta :=min\{\frac{s}{2s+2\beta \left(\alpha \right)+1},\frac{s-1/r+1/p}{2(s-1/r)+2\beta \left(\alpha \right)+1}\}=\frac{s}{2s+2\beta \left(\alpha \right)+1}$. It is easy to see that $\frac{|{\beta }_{jk}|}{{\tau }_{j,n}/2}>1$ for $k\in B_j$. Hence,

$$\begin{array}{c}\hfill E{e}_{22}\lesssim \sum _{j=j_0^{\ast }+1}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}E{|{\xi }_{jk}|}^p{\left(\frac{|{\beta }_{jk}|}{{\tau }_{j,n}/2}\right)}^r.\end{array}$$

(18)

In addition, $E|{\xi }_{jk}{|}^p\lesssim 2^{j\beta \left(\alpha \right)p}{n}^{-\frac{p}{2}}$ by Lemma 2 and ${\sum }_k{|{\beta }_{jk}|}^r\lesssim 2^{-j(sr+\frac{r}{2}-1)}$ due to Proposition 1 and $f\in B_{r,q}^s$. These with (18), ${\tau }_{j,n}\sim 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$ and $2^{j_0^{\ast }}\sim {\left(\frac{n}{lnn}\right)}^{\frac{1-2\theta }{2\beta \left(\alpha \right)+1}}$ lead to

$$\begin{array}{c}\hfill E{e}_{22}\lesssim \sum _{j=j_0^{\ast }+1}^{j_1}{\left(\frac{lnn}{n}\right)}^{\frac{p-r}{2}}{2}^{-j\omega }\lesssim {\left(\frac{lnn}{n}\right)}^{\frac{p-r}{2}}{2}^{-j_0^{\ast }\omega }\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}.\end{array}$$

For the case $\omega <0$, $\theta :=min\{\frac{s}{2s+2\beta \left(\alpha \right)+1},\frac{s-1/r+1/p}{2(s-1/r)+2\beta \left(\alpha \right)+1}\}=\frac{s-1/r+1/p}{2(s-1/r)+2\beta \left(\alpha \right)+1}$. Denote $r_1:=(1-2\theta )p$. Then $r\le r_1\le p$. Hence, the same arguments as (18) show that

$$\begin{array}{c}\hfill E{e}_{22}\lesssim \sum _{j=j_0^{\ast }+1}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}E{|{\xi }_{jk}|}^p{\left(\frac{|{\beta }_{jk}|}{{\tau }_{j,n}/2}\right)}^{r_1}.\end{array}$$

(19)

On the other hand, ${\sum }_k|{\beta }_{jk}{|}^{r_1}\lesssim ({\sum }_k|{\beta }_{jk}{{|}^r)}^{\frac{r_1}{r}}\lesssim 2^{-j{r}_1(s+\frac{1}{2}-\frac{1}{r})}$ follows from $f\in B_{r,q}^s$ and Proposition 1. Furthermore, $E|{\xi }_{jk}{|}^p\lesssim 2^{j\beta \left(\alpha \right)p}{n}^{-\frac{p}{2}}$ thanks to Lemma 2. These with (19) and ${\tau }_{j,n}\sim 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$ imply that

$$\begin{array}{c}\hfill E{e}_{22}\lesssim \sum _{j=j_0^{\ast }+1}^{j_1}{2}^{j[\frac{p}{2}-1+\beta \left(\alpha \right)p-\beta \left(\alpha \right)r_1-r_1(s+\frac{1}{2}-\frac{1}{r})]}{\left(\frac{lnn}{n}\right)}^{\frac{p-r_1}{2}}\lesssim lnn{\left(\frac{lnn}{n}\right)}^{\theta p},\end{array}$$

because $\frac{p-r_1}{2}=\theta p$ and $\frac{p}{2}-1+\beta \left(\alpha \right)p-\beta \left(\alpha \right)r_1-r_1(s+\frac{1}{2}-\frac{1}{r})=0$.

Finally, it remains to estimate $E{e}_4$. Obviously, $\theta =\frac{s}{2s+2\beta \left(\alpha \right)+1}$ for $\omega \ge 0$. Thus,

$$\begin{array}{c}\hfill e_4=(\sum _{j=j_0}^{j_0^{\ast }}+\sum _{j=j_0^{\ast }+1}^{j_1})2^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\beta }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j^c\cap C_j^c\}}:=e_{41}+e_{42}.\end{array}$$

(20)

Because $k\in C_j^c$ and ${\tau }_{j,n}\sim 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$, $|{\beta }_{jk}|<2{\tau }_{j,n}\sim 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$. This with $|{\Lambda }_j|\sim 2^j$ implies that

$$\begin{array}{c}\hfill E{e}_{41}\lesssim \sum _{j=j_0}^{j_0^{\ast }}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{\tau }_{j,n}^p\lesssim 2^{j_0^{\ast }p(\beta \left(\alpha \right)+\frac{1}{2})}{\left(\frac{lnn}{n}\right)}^{\frac{p}{2}}.\end{array}$$

(21)

Note that $\frac{2{\tau }_{j,n}}{|{\beta }_{jk}|}>1$ by $k\in C_j^c$, and ${\sum }_k{|{\beta }_{jk}|}^r\lesssim 2^{-j(sr+\frac{r}{2}-1)}$ due to $f\in B_{r,q}^s$. Subsequently, with $r\le p$ and ${\tau }_{j,n}\sim 2^{j\beta \left(\alpha \right)}\sqrt{\frac{j}{n}}$,

$$\begin{array}{c}\hfill E{e}_{42}\lesssim \sum _{j=j_0^{\ast }+1}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _k{|{\beta }_{jk}|}^p{\left(\frac{2{\tau }_{j,n}}{|{\beta }_{jk}|}\right)}^{p-r}\lesssim 2^{-j_0^{\ast }\omega }{\left(\frac{lnn}{n}\right)}^{\frac{p-r}{2}}.\end{array}$$

(22)

Combining (20)–(22) with $2^{j_0^{\ast }}\sim {\left(\frac{n}{lnn}\right)}^{\frac{1-2\theta }{2\beta \left(\alpha \right)+1}}$ and $\theta =\frac{s}{2s+2\beta \left(\alpha \right)+1}(\omega \ge 0)$, one obtains that

$$E{e}_4=E{e}_{41}+E{e}_{42}\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}$$

holds for $\omega \ge 0$.

At last, one shows $E{e}_4\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}$ for the case $\omega <0.$ Subsequently,

$$\begin{array}{c}\hfill e_4=(\sum _{j=j_0}^{j_1^{\ast }}+\sum _{j=j_1^{\ast }+1}^{j_1})2^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\beta }_{jk}|}^p{I}_{\{k\in {\widehat{B}}_j^c\cap C_j^c\}}:=e_{41}^{\prime }+e_{42}^{\prime }.\end{array}$$

(23)

The same arguments as (22), one finds

$$\begin{array}{c}\hfill E{e}_{41}^{\prime }\lesssim 2^{-j_1^{\ast }\omega }{\left(\frac{lnn}{n}\right)}^{\frac{p-r}{2}}.\end{array}$$

(24)

On the other hand, ${\sum }_k|{\beta }_{jk}{|}^p\lesssim ({\sum }_k|{\beta }_{jk}{{|}^r)}^{\frac{p}{r}}\lesssim 2^{-jp(s+\frac{1}{2}-\frac{1}{r})}$ by $f\in B_{r,q}^s(r\le p)$ and Proposition 1. Furthermore,

$$\begin{array}{c}\hfill E{e}_{42}^{\prime }\le \sum _{j=j_1^{\ast }+1}^{j_1}{2}^{j(\frac{p}{2}-1)}\sum _{k\in {\Lambda }_j}{|{\beta }_{jk}|}^p\lesssim 2^{-j_1^{\ast }p(s-\frac{1}{r}+\frac{1}{p})}\end{array}$$

because of $s>\frac{1}{r}$. This with (23) and (24) and $2^{j_1^{\ast }}\sim {\left(\frac{n}{lnn}\right)}^{\frac{\theta }{s-\frac{1}{r}+\frac{1}{p}}}$ leads to

$$E{e}_4=E{e}_{41}^{\prime }+E{e}_{42}^{\prime }\lesssim {\left(\frac{lnn}{n}\right)}^{\theta p}$$

thanks to $\theta =\frac{s-1/r+1/p}{2(s-1/r)+2\beta \left(\alpha \right)+1}$ for $\omega <0.$ The proof is completed. □

4. Conclusions

This current paper shows $L^p(1\le p<\infty )$ risk estimations of both linear and non-linear wavelet estimators under a convolution structure density model over Besov balls. The corresponding conclusions are introduced by Theorems 1 and 2 in this paper, which can be seen as an extension of the works of Donoho et al. [1] and Li & Liu [4].

It should be pointed out that the non-linear wavelet estimator is adaptive, and the convergence rate of non-linear estimator are better than that of linear one for the case $r\le p$. However, it is not clear whether the estimations are optimal (nearly-optimal) or not for $\alpha \in (0,1)$. Therefore, one of our future work is to determine a low bound estimate for model (2) with $\alpha \in (0,1)$.