Adaptive wavelet estimations for the derivative of a density in GARCH-type model

Abstract

Recently, Rao investigated the estimations for the derivative of a density in GARCH-type model $S=\sigma ^{2}Z$ over $L^{2}$-risk (Commun. Stat., Theory Methods 46:2396–2410, 2017). This paper extends those estimations to $L^{p}$-risk ($1\leq p<\infty$). In addition, we provide a lower bound for this model, which indicates one of our convergence rates to be nearly-optimal.

1 Introduction

The GARCH-type model

$$\begin{aligned} S=\sigma ^{2}Z \end{aligned}$$

is considered in this paper, where $\sigma ^{2}$ and Z are independent random variables. In practice, we always assume that the density function $f_{\sigma ^{2}}$ of $\sigma ^{2}$ is unknown and $\operatorname{supp} f_{\sigma ^{2}}\subseteq [0,1]$, while the density of Z is known. We want to estimate the first derivative of $f_{\sigma ^{2}}$ based on n independent and identically distributed (i.i.d.) observed samples $S_{1},\ldots ,S_{n}$ of S by wavelet methods, so that we also need suppose the differentiability of $f_{\sigma ^{2}}$ and $f'_{\sigma ^{2}}\in L^{p}([0,1])$.

Non-parametric estimations of a density and regression function are widely investigated in the literature [12, 14, 16]. It is well known that the estimations for the derivatives of a density are also important and interesting, which could reflect monotonicity, concavity or convexity properties of density functions. Asymptotic properties of the kernel estimators for a density derivative have been considered earlier in [15], while the wavelet type estimator was discussed in [17].

As usual, we consider the $L^{p}$ minimax risk ($L^{p}$-risk) [13],

$$\begin{aligned} \inf_{\hat{f}_{n}}\sup_{f_{\sigma ^{2}}\in \varSigma } E \Vert \hat{f_{n}}-f _{\sigma ^{2}} \Vert _{p}, \end{aligned}$$

where the infimum runs over all possible estimators $\hat{f}_{n}$ and Σ is a class of functions. Here and after, EX stands for the mathematical expectation of a random variable X and $\|f\|_{p}$ denotes the ordinary $L^{p}$ norm.

In 2012, Chesneau and Doosti [9] investigated the wavelet estimation of density for GARCH model under various dependence structures. Next year, Chesneau [8] studied the wavelet estimation of a density in GARCH-type model leading to upper bounds under $L^{2}$-risk. In 2017, Rao [17] considered $L^{2}$-risk for the derivative of a density in GARCH-type model over a Besov ball by wavelets.

In this paper, we address to extend Rao’s work [17] to $L^{p}$-risk $(1\leq p<\infty )$. Moreover, we show that one of our convergence rates is nearly-optimal. On the other hand, this work can also be seen as a generalization of multiplicative censoring model. Vardi [18, 19] introduced the multiplicative censoring model which unifies several models including non-parametric inference for renewal processes, non-parametric deconvolution problems and estimation of decreasing density functions. Recently, Abbaszadeh et al. [1] considered the wavelet estimation of a density and its derivatives under $L^{p}$-risk ($1\leq p<\infty$) in the multiplicative censoring one. The density estimations for the multiplicative censoring model also can be found in [2, 3] and [6, 7].

This paper is organized as follows. Section 2 briefly describes the Besov ball and wavelet estimators. The theoretical results are given in Sect. 3. Some lemmas are provided in Sect. 4. The proofs are gathered in Sect. 5.

2 Besov ball and estimators

This section describes the Besov ball and wavelet estimators. First, we introduce the Besov ball and its wavelet characterizations.

2.1 Besov ball

Let $W_{r}^{n}(\mathbb{R})$ be the Sobolev space with a non-negative integer n,

$$\begin{aligned} W_{r}^{n}(\mathbb{R}):=\bigl\{ f: f\in L^{r}( \mathbb{R}), f^{(n)}\in L^{r}( \mathbb{R})\bigr\} , \end{aligned}$$

and $\|f\|_{W_{r}^{n}}:=\|f\|_{r}+\|f^{(n)}\|_{r}$. Then $L^{r}( \mathbb{R})$ can be considered as $W_{r}^{0}(\mathbb{R})$. For $1\leq r,q\leq \infty$ and $s=n+\alpha$ with $\alpha \in (0,1]$, a Besov space $B_{r,q}^{s}(\mathbb{R})$ is defined by

$$\begin{aligned} B_{r,q}^{s}(\mathbb{R}):=\bigl\{ f: f\in W_{r}^{n}( \mathbb{R}), \bigl\Vert t^{- \alpha }\omega _{r}^{2} \bigl(f^{(n)},t\bigr) \bigr\Vert _{q}^{\ast }< \infty \bigr\} \end{aligned}$$

with the norm $\|f\|_{B_{r,q}^{s}}:=\|f\|_{W_{r}^{n}}+\|t^{-\alpha } \omega _{r}^{2}(f^{(n)},t)\|_{q}^{\ast }$. Here, $\omega _{r}^{2}(f,t):= \sup_{|h|\leq t}\|f(\cdot +2h)-2f(\cdot +h)+f(\cdot )\|_{r}$ denotes the smoothness modulus of f and

$$\begin{aligned} \Vert h \Vert _{q}^{\ast }:= \textstyle\begin{cases} (\int _{0}^{+\infty } \vert h(t) \vert ^{q}\frac{dt}{t})^{\frac{1}{q}} & \text{if $1\leq q< \infty $;} \\ \mathop{\mathrm{ess sup}} _{t} \vert h(t) \vert & \text{if $q=\infty $.} \end{cases}\displaystyle \end{aligned}$$

When $s>0$ and $1\leq r,q,r' \leq \infty$, it is well known that

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

• (ii) $B_{r,q}^{s}\hookrightarrow B_{r',q}^{s'}$ for $r\leq r'$ and $s-\frac{1}{r}=s'-\frac{1}{r'}$;

• (iii) $B_{\infty ,\infty }^{s}(\mathbb{R})$ is the classical Hölder space $H^{s} (\mathbb{R})$,

where $A\hookrightarrow B$ stands for a Banach space A continuously embedded in another Banach space B. More precisely, $\|u\|_{B} \leq c_{1}\|u\|_{A}$ ($u\in A$) holds for some constant $c_{1}>0$. By (i), $B_{r,q}^{s}(\mathbb{R})\hookrightarrow L^{\infty }(\mathbb{R})$ for $s>\frac{1}{r}$. All these notations and claims can be found in [13].

In this paper, a Besov ball

$$\begin{aligned} B_{r,q}^{s}(M)=\bigl\{ f\in B_{r,q}^{s}( \mathbb{R}): \Vert f \Vert _{B_{r,q}^{s}} \leq M\bigr\} ,\quad M>0, \end{aligned}$$

is considered.

Let ϕ be a scaling function and ψ be the corresponding wavelet function such that

$$\begin{aligned} \{\phi _{\tau ,k}, \psi _{j,k}: j\geq \tau , k\in \mathbb{Z}\} \end{aligned}$$

constitutes an orthonormal basis of $L^{2}(\mathbb{R})$, where τ is a positive integer and $g_{j,k}(x)=2^{\frac{j}{2}}g(2^{j}x-k)$ for $g=\phi$ or ψ. Then, for $h\in L^{2}(\mathbb{R})$,

$$\begin{aligned} h=\sum_{k\in \varOmega _{\tau }}\alpha _{\tau ,k} \phi _{\tau ,k} +\sum_{j= \tau }^{\infty }\sum _{k\in \varOmega _{j}}\beta _{j,k}\psi _{j,k} \end{aligned}$$

(1)

with $\alpha _{j,k}=\langle h,\phi _{j,k}\rangle$, $\beta _{j,k}=\langle h,\psi _{j,k}\rangle$ and

$$\begin{aligned} \varOmega _{j}=\{k\in \mathbb{Z}:\operatorname{supp} h\cap \operatorname{supp} \phi _{j,k}\neq \emptyset \}\cup \{k\in \mathbb{Z}:\operatorname{supp} h\cap \operatorname{supp} \psi _{j,k}\neq \emptyset \}. \end{aligned}$$

In particular, when $\phi ,\psi$ and h have compact supports, the cardinality of $\varOmega _{j}$ satisfies $|\varOmega _{j}|\leq C2^{j}$, where $C>0$ is a constant depending only on the support lengths of $\phi , \psi$ and h.

As usual, the orthogonal projection operator $P_{j}$ is given by

$$\begin{aligned} P_{j}h=\sum_{k\in \varOmega _{j}}\alpha _{j,k}\phi _{j,k}. \end{aligned}$$

(2)

When $\phi \in C^{m}$ (so does ψ) is compactly supported, the identity (1) and (2) hold in $L^{p}$ sense for $p\geq 1$ [13]. Here and throughout, $C^{m}$ stands for the set consisting of all m times continuously differentiable functions.

The following wavelet characterization theorem of Besov space is needed in Sect. 5.

Lemma 2.1

([13])

Let a scaling function $\phi \in C^{m}$ be compactly supported. Then, for $r,q\in [1,+\infty ], 0< s< m$ and $h\in L^{r}(\mathbb{R})$, the following assertions are equivalent:

$$\begin{aligned} &\mathrm{(i)}\quad h\in B_{r,q}^{s}(\mathbb{R}); \qquad \mathrm{(ii)}\quad 2^{js} \Vert P_{j}h-h \Vert _{r}\in l_{q}; \\ &\mathrm{(iii)} \quad \Vert \alpha _{j_{0},\cdot } \Vert _{l_{r}}+ \bigl\Vert \bigl\{ 2^{j(s+\frac{1}{2} - \frac{1}{r})} \Vert \beta _{j_{0},\cdot } \Vert _{l_{r}} \bigr\} _{j\geq j_{0}} \bigr\Vert _{l _{q}}< \infty. \end{aligned}$$

In each case,

$$\begin{aligned} \Vert h \Vert _{B_{r,q}^{s}}\thicksim \Vert h \Vert _{s,r,q}:= \Vert \alpha _{j_{0},\cdot } \Vert _{l_{r}}+ \bigl\Vert \bigl\{ 2^{j(s+\frac{1}{2} -\frac{1}{r})} \Vert \beta _{j_{0},\cdot } \Vert _{l_{r}}\bigr\} _{j\geq j_{0}} \bigr\Vert _{l_{q}}. \end{aligned}$$

Here and afterwards, $A\lesssim B$ means $A\leq c_{2}B$ for some constant $c_{2}>0$; $A\gtrsim B$ denotes $B\lesssim A$; we also use $A\thicksim B$ to stand for both $A\lesssim B$ and $A\gtrsim B$.

2.2 Estimators

This part introduces our wavelet estimators for the GARCH-type model $S=\sigma ^{2}Z$ described earlier. Suppose

$$\begin{aligned} Z=\prod_{i=1}^{v}U_{i}, \end{aligned}$$

where v is a known positive integer and $U_{1},\ldots ,U_{v}$ are i.i.d. random variables with standard uniform distribution. Clearly, the density function of Z satisfies

$$\begin{aligned} f_{Z}(z)=\frac{1}{(v-1)!}(-\ln z)^{v-1},\quad 0\leq z\leq 1. \end{aligned}$$

As in [8, 17], we assume that there exists a known constant $C_{\ast }$ such that

$$\begin{aligned} \sup_{x\in [0,1]}f_{s}(x)\leq C_{\ast }, \end{aligned}$$

(3)

where $f_{s}$ is the density function of S.

For any $x\in [0,1]$, $h\in C^{k}([0,1])$, we define

$$\begin{aligned} T(h) (x)=\bigl(xh(x)\bigr)'=h(x)+xh'(x),\qquad T_{k}(h) (x)=T\bigl(T_{k-1}(h)\bigr) (x) \end{aligned}$$

(4)

and

$$\begin{aligned} G(h) (x)=-xh'(x),\qquad G_{k}(h) (x)=G\bigl(G_{k-1}(h)\bigr) (x), \end{aligned}$$

(5)

where k is a positive integer. Then the following lemma holds.

Lemma 2.2

([8])

Let G and T be defined as above. Then

• (i) $f_{{\sigma }^{2}}(x)=G_{v}(f_{s})(x), x\in [0,1]$;

• (ii) For any $h\in C^{v}([0,1])$,

  $$\begin{aligned} \int _{0}^{1}f_{{\sigma }^{2}}(x)h(x)\,dx= \int _{0}^{1}f_{s}(x)T_{v}(h) (x)\,dx. \end{aligned}$$

Next, we will introduce wavelet estimators, which can be found in Ref. [17]. Define

$$\begin{aligned} \widehat{\alpha }_{j_{0},k}=-{\frac{1}{n}} {\sum _{i=1}^{n}} T _{v} \bigl(({\phi }_{j,k})'\bigr) (S_{i}) \quad\text{and}\quad \widehat{\beta }_{j,k}=-{\frac{1}{n}} {\sum _{i=1}^{n}} T_{v} \bigl(( {\psi }_{j,k})'\bigr) (S_{i}). \end{aligned}$$

(6)

Here and after, let ϕ be Daubechies’ scaling function $D_{2N}$ with large N and ψ be the corresponding wavelet function. It is well known that $\phi ,\psi \in C^{v+1}$ with N large enough. Furthermore, the linear wavelet estimator is given by

$$\begin{aligned} \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}= \sum _{k\in \varOmega _{j_{0}}} \widehat{\alpha }_{j_{0},k}{\phi }_{j_{0},k}, \end{aligned}$$

(7)

where $j_{0}$ is a positive integer which will be chosen later.

In order to get adaptivity, we need the thresholding method [4, 14, 17]. As in [17], let

$$\begin{aligned} 2^{j_{1}}\thicksim \biggl(\frac{n}{\ln n}\biggr)^{\frac{1}{2(v+1)+1}},\qquad \lambda _{j}=2^{(v+1)j}\sqrt{\frac{j}{n}},\qquad \widetilde{\beta }_{j,k}= \widehat{\beta }_{j,k}I\bigl\{ \vert \widehat{ \beta }_{j,k} \vert \geq \varUpsilon \lambda _{j}\bigr\} \end{aligned}$$

with the constants $\varUpsilon =c\gamma$, $c>\max \{8C_{\min },1\}$ and $\gamma \geq p(2v+3)$. Here,

$$\begin{aligned} C_{\min }=(v+2)!\sum_{u=0}^{v} \bigl[(v+1) (v+2)!C_{\ast } \bigl\Vert \psi ^{(u+1)} \bigr\Vert _{2}^{2}+2 \bigl\Vert \psi ^{(u+1)} \bigr\Vert _{\infty } \bigr] \end{aligned}$$

(8)

with $C_{\ast }$ given in (3). This special choice c is used in Lemma 4.3, while $\gamma \geq p(2v+3)$ is needed in the estimations of $Ee_{1}$ and $Ee_{3}$ (see Sect. 5). Here, we replace $\lambda _{j}=2^{(v+1)j}\sqrt{\frac{\ln n}{n}}$ (see [17]) by $\lambda _{j}=2^{(v+1)j}\sqrt{\frac{j}{n}}$, which is used in the proof of Lemma 4.3. In fact, the universal threshold of classical adaptive density estimation is $\sqrt{ \frac{j}{n}}$ (see [11]) and two forms do not influence the convergence rates of our results.

The nonlinear wavelet estimator is given by

$$\begin{aligned} \widehat{f'_{\sigma ^{2}}}^{\mathrm{non}}=\sum _{k\in \varOmega _{\tau }} \widehat{\alpha }_{\tau ,k} {\phi }_{\tau ,k}+{\sum_{j={\tau }} ^{j_{1}}} \sum _{k\in \varOmega _{j}}\widetilde{\beta } _{j,k} {\psi }_{j,k} \end{aligned}$$

(9)

with some positive integer τ.

3 Results

This section describes the results in this paper.

Theorem 3.1

Assume $r\in [1,+\infty ), q\in [1,+\infty ]$ and $s>\frac{1}{r}$, then, for $p\in [1,+\infty )$, the estimator $\widehat{f'_{{\sigma }^{2}}} ^{\mathrm{lin}}$ in (7) with $2^{j_{0}}\sim n^{ \frac{1}{2s'+2(v+1)+1}}$ satisfies

$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}, \end{aligned}$$

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

Remark 1

When $p=2$ and $r\geq 2$, the above estimation shows

$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{2}^{2} \lesssim n^{-\frac{2s}{2s+2(v+1)+1}}, \end{aligned}$$

which coincides with Theorem 5.1 of Ref. [17].

Remark 2

The condition $s>\frac{1}{r}$ can be replaced by $s'=s-(\frac{1}{r}- \frac{1}{p})_{+}>0$, because the former condition is only used to conclude $B_{r,q}^{s}\hookrightarrow B_{p,q}^{s'}$ in the proof of Theorem 3.1.

The next theorem gives an adaptive upper bound estimation by the nonlinear wavelet estimator $\widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}$ in (9).

Theorem 3.2

Let $r\in [1,+\infty ), q\in [1,+\infty ]$ and $s>\frac{1}{r}$. Then, for $p\in [1,+\infty )$,

$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim (\ln n)^{p} \bigl(n^{-1}\ln n\bigr)^{\alpha p} \end{aligned}$$

with $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}$.

Remark 3

When $sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p\geq 0$, $\alpha = \frac{s}{2s+2(v+1)+1}$. In particular, the above result with $p=2$ coincides with Theorem 5.2 in [17].

Remark 4

The condition $s>\frac{1}{r}$ in Theorem 3.2 can’t be replaced by $s'=s- (\frac{1}{r}-\frac{1}{p})_{+}>0$ for $r\leq p$, since we need $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}\leq \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\leq \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(v+1)+1}$ for the estimation of $A_{3}$ in Sect. 5.

Remark 5

Let m be a constant such that $m>s$, and $2^{j_{0}}\thicksim n^{ \frac{1}{2m+2(v+1)+1}}$. Then the number of calculations can be reduced effectively, when the level τ in $\widehat{f'_{{\sigma }^{2}}} ^{\mathrm{non}}$ is replaced by $j_{0}$.

The following theorem shows a lower bound estimation.

Theorem 3.3

Assume $s>0$ and $r,q\in [1,+\infty ]$, then, for any $p\in [1,+ \infty )$,

$$\begin{aligned} \inf_{\widehat{f}'_{{\sigma }^{2}}}\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f}'_{{\sigma } ^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r}+ \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}, \end{aligned}$$

where $\widehat{f}'_{{\sigma }^{2}}$ runs over all possible estimators of $f'_{{\sigma }^{2}}$.

Remark 6

Combining Theorem 3.3 with Theorem 3.2, we find that the convergence rate $\frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}$ is nearly-optimal. As for the other one, we will study it below.

4 Some lemmas

This section is devoted to providing some lemmas, which are needed for the proofs of our theorems.

Lemma 4.1

([13])

Let g be a scaling function or a wavelet function with

$$\begin{aligned} \sup_{x\in \mathbb{R}}\sum_{k} \bigl\vert g(x-k) \bigr\vert < +\infty. \end{aligned}$$

Then there exists $C>0$ such that, for $\lambda =\{\lambda _{k}\} \in l^{p}(\mathbb{Z})$ and $1\leq p\leq \infty$,

$$\begin{aligned} \biggl\Vert \sum_{k\in {\mathbb{Z}}}\lambda _{k}g_{jk} \biggr\Vert _{p}\leq C2^{j( \frac{1}{2}-\frac{1}{p})} \Vert \lambda \Vert _{l_{p}}. \end{aligned}$$

We need the well-known Rosenthal’s inequality [13], in order to prove Lemma 4.2.

Rosenthal’s inequality. Let $X_{1},\ldots ,X_{n}$ be independent random variables and $EX_{i}=0$. Then

$$\begin{aligned} E \Biggl\vert \sum_{i=1}^{n}X_{i} \Biggr\vert ^{p}\leq \textstyle\begin{cases} C_{p} [ \sum_{i=1}^{n}E \vert X_{i} \vert ^{p}+ (\sum_{i=1}^{n}E \vert X_{i} \vert ^{2} )^{\frac{p}{2}} ], &2\leq p< \infty; \\ (\sum_{i=1}^{n}E \vert X_{i} \vert ^{2} )^{\frac{p}{2}}, &0< p< 2, \end{cases}\displaystyle \end{aligned}$$

where $C_{p}>0$ is a constant.

Lemma 4.2

Let $\widehat{\alpha }_{j,k}$ and $\widehat{\beta }_{j,k}$ be given by (6). Then, for $p\in (0,+\infty )$,

• (i) $E\widehat{\alpha }_{j,k}=\alpha _{j,k}, E\widehat{\beta }_{j,k}= \beta _{j,k}$;

• (ii) $E|\widehat{\alpha }_{j,k}- {\alpha }_{j,k}|^{p} \lesssim n^{- \frac{p}{2}} 2^{(v+1)jp}, E|\widehat{\beta }_{j,k}- {\beta }_{j,k}|^{p} \lesssim n^{-\frac{p}{2}} 2^{(v+1)jp}$,

where $\alpha _{j,k}=\langle f'_{\sigma ^{2}}, \phi _{j,k}\rangle$ and $\beta _{j,k}=\langle f'_{\sigma ^{2}}, \psi _{j,k}\rangle$.

Proof

(i) One only need prove $E\widehat{\alpha }_{j,k}={\alpha }_{j,k}$ and the second one is the same. According to the definition of $\widehat{\alpha }_{j,k}$ in (6), one gets

$$\begin{aligned} E\widehat{\alpha }_{j,k}=-E\bigl[T_{v} \bigl(({\phi }_{j,k})'\bigr) (S_{1})\bigr]=- \int _{0}^{1} T_{v} \bigl(({\phi }_{j,k})'\bigr) (x)f_{s}(x)\,dx \end{aligned}$$

thanks to $S_{1},\ldots ,S_{n}$ are i.i.d. On the other hand, $f_{{\sigma }^{2}}(0)=f_{{\sigma }^{2}}(1)=0$ follows from $\operatorname{supp} f_{{\sigma }^{2}}\subseteq [0,1]$ and the continuity of $f_{{\sigma }^{2}}$. These with Lemma 2.2 imply

$$ E\widehat{\alpha }_{j,k}=- \int _{0}^{1}f_{{\sigma }^{2}}(x) ({\phi } _{j,k})'(x)\,dx=-f_{{\sigma }^{2}}(x){\phi }_{j,k}(x)| _{0}^{1} + \int _{0}^{1}f'_{{\sigma }^{2}}(x){\phi }_{j,k}(x)\,dx={\alpha }_{j,k}. $$

(ii) One also prove the first inequality and the second one is similar. By (6) and the results of (i),

$$\begin{aligned} \widehat{\alpha }_{j,k}-{\alpha }_{j,k}= \widehat{\alpha }_{j,k}-E \widehat{\alpha }_{j,k}= {\frac{1}{n}} {\sum _{i=1}^{n}} \bigl\{ E\bigl[T _{v} \bigl(({\phi }_{j,k})'\bigr) (S_{i}) \bigr]-T_{v} \bigl(({\phi }_{j,k})'\bigr) (S_{i}) \bigr\} . \end{aligned}$$

Let $X_{i}:=E[T_{v} (({\phi }_{j,k})')(S_{i})]-T_{v} (({\phi }_{j,k})')(S _{i})$. Then $X_{1},\ldots ,X_{n}$ are i.i.d., $EX_{i}=0$ and

$$\begin{aligned} E \vert \widehat{\alpha }_{j,k}-{\alpha }_{j,k} \vert ^{p}= E \Biggl\vert {\frac{1}{n}} { \sum_{i=1}^{n}}X_{i} \Biggr\vert ^{p}= n^{-p}E \Biggl\vert {\sum _{i=1} ^{n}}X_{i} \Biggr\vert ^{p}. \end{aligned}$$

(10)

According to (4),

$$\begin{aligned} \bigl|T_{v}\bigl(({\phi }_{j,k})' \bigr) (x)\bigr|\leq (v+2)!\sum_{u=0}^{v} \bigl\vert x^{u}(\phi _{j,k})^{(u+1)}(x) \bigr\vert . \end{aligned}$$

(11)

Hence,

$$\begin{aligned} \sup_{x\in [0,1]}\bigl|T_{v} \bigl(({\phi }_{j,k})' \bigr) (x)\bigr| &\leq (v+2)! \sup_{x\in [0,1]}\sum _{u=0}^{v} \bigl\vert x^{u} (\phi _{j,k})^{(u+1)}(x) \bigr\vert \\ &\leq (v+2)!\sum_{u=0}^{v}\sup _{x\in [0,1]} \bigl\vert (\phi _{j,k})^{(u+1)}(x) \bigr\vert \\ &\leq (v+2)! \sum_{u=0}^{v} \bigl\Vert \phi ^{(u+1)} \bigr\Vert _{\infty }2^{(v+ \frac{3}{2})j}. \end{aligned}$$

Clearly,

$$\begin{aligned} \vert X_{i} \vert \leq E\bigl|T_{v} \bigl(({\phi }_{j,k})'\bigr) (S_{i}) \bigr| +\bigl| T_{v}\bigl(({\phi }_{j,k})'\bigr) (S _{i})\bigr|\leq C_{1} 2^{(v+\frac{3}{2})j}, \end{aligned}$$

(12)

where $C_{1}=2(v+2)!\sum_{u=0}^{v}\|\phi ^{(u+1)}\|_{\infty }$. On the other hand, $\sup_{x\in [0,1]}f_{s}(x)\leq C_{\ast }$ in (3) and $S_{i}\in [0,1]$ show that

$$\begin{aligned} E \bigl\vert (\phi _{j,k})^{(u+1)}(S_{i}) \bigr\vert ^{2}= \int _{0}^{1} \bigl\vert (\phi _{j,k})^{(u+1)}(x) \bigr\vert ^{2}f _{s}(x)\,dx\leq C_{\ast } \bigl\Vert \phi ^{(u+1)} \bigr\Vert _{2}^{2}2^{(2u+2)j}. \end{aligned}$$

This with (11) and $S_{i}\in [0,1]$ leads to

$$\begin{aligned} E \bigl[T_{v} \bigl(({\phi }_{j,k})'\bigr) (S_{i}) \bigr]^{2} &\leq \bigl[(v+2)!\bigr]^{2}E \Biggl[\sum_{u=0}^{v} \bigl\vert S_{i}^{u}(\phi _{j,k})^{(u+1)}(S_{i}) \bigr\vert \Biggr]^{2} \\ &\leq (v+1)\bigl[(v+2)!\bigr]^{2}\sum_{u=0}^{v}E \bigl\vert (\phi _{j,k})^{(u+1)}(S _{i}) \bigr\vert ^{2} \\ &\leq (v+1)\bigl[(v+2)!\bigr]^{2}C_{\ast }\sum _{u=0}^{v} \bigl\Vert \phi ^{(u+1)} \bigr\Vert _{2}^{2}2^{(2v+2)j}. \end{aligned}$$

Furthermore,

$$\begin{aligned} E \vert X_{i} \vert ^{2}\leq E \bigl[ T_{v} \bigl(({\phi }_{j,k})'\bigr) (S_{i}) \bigr]^{2} \leq C_{2}2^{(2v+2)j}, \end{aligned}$$

(13)

where $C_{2}=(v+1)[(v+2)!]^{2}C_{\ast }\sum_{u=0}^{v}\|\phi ^{(u+1)}\|_{2}^{2}$.

When $0< p<2$, by using (10), Jensen’s inequality and (13),

$$\begin{aligned} E \vert \widehat{\alpha }_{j,k}-{\alpha }_{j,k} \vert ^{p}= n^{-p}E \Biggl\vert {\sum _{i=1}^{n}}X_{i} \Biggr\vert ^{p} \lesssim n^{-p} \Biggl[\sum _{i=1} ^{n}E \vert X_{i} \vert ^{2} \Biggr]^{\frac{p}{2}}\lesssim n^{-\frac{p}{2}}2^{(v+1)jp}. \end{aligned}$$

For the case of $2\leq p<\infty$, according to Rosenthal’s inequality,

$$\begin{aligned} E \Biggl\vert \sum_{i=1}^{n} X_{i} \Biggr\vert ^{p} &\lesssim \sum _{i=1}^{n}E \vert X_{i} \vert ^{p}+ \Biggl(\sum_{i=1}^{n}E \vert X_{i} \vert ^{2} \Biggr)^{ \frac{p}{2}} \\ &\lesssim n2^{(v+\frac{3}{2})(p-2)j}2^{(2v+2)j}+\bigl(n2^{(2v+2)j} \bigr)^{ \frac{p}{2}} \\ &\lesssim n^{\frac{p}{2}}2^{(v+1)pj}\bigl[n^{1-\frac{p}{2}}2^{( \frac{p}{2}-1)j}+1 \bigr] \end{aligned}$$

because of (12) and (13). Moreover, $n^{1- \frac{p}{2}} 2^{(\frac{p}{2}-1)j}\leq 1$ follows from $2^{j}\leq n$ and $p\geq 2$. Then

$$\begin{aligned} E \vert \widehat{\alpha }_{j,k}-{\alpha }_{j,k} \vert ^{p}= n^{-p}E \Biggl\vert {\sum _{i=1}^{n}}X_{i} \Biggr\vert ^{p}\lesssim n^{-\frac{p}{2}}2^{(v+1)pj} \end{aligned}$$

due to (10). This completes the proof. □

Bernstein’s inequality [13] is necessary in the proof of Lemma 4.3.

Bernstein’s inequality. Let $X_{1},\ldots ,X_{n}$ be i.i.d. random variables, $EX_{i}=0$ and $|X_{i}|\leq \|X\|_{\infty }$ ($i=1, \ldots ,n$). Then, for each $\gamma >0$,

$$\begin{aligned} P \Biggl\{ \Biggl\vert \frac{1}{n}\sum_{i=1}^{n} \Biggr\vert >\gamma \Biggr\} \leq 2 \exp \biggl(-\frac{n\gamma ^{2}}{2(EX_{i}^{2}+ \Vert X \Vert _{\infty }\gamma /3)} \biggr). \end{aligned}$$

Lemma 4.3

Let $\beta _{j,k}$ be the wavelet coefficient of $f'_{\sigma ^{2}}$, $\widehat{\beta }_{j,k}$ be defined in (6) and $\varUpsilon =c\gamma$. Then, for any $j>0$, $j2^{j}\leq n$ and $\gamma \geq 1$, there exists a constant $c\geq \max \{8C_{\min },1\}$ such that

$$\begin{aligned} P\bigl\{ \vert \widehat{\beta }_{j,k}- {\beta }_{j,k} \vert >\varUpsilon \lambda _{j}/2\bigr\} \lesssim 2^{-\gamma j}, \end{aligned}$$

where $C_{\min }$ is given by (8).

Proof

According to the definition of $\widehat{\beta }_{j,k}$ in (6), one obtains

$$ \widehat{\beta }_{j,k}-{\beta }_{j,k} =\frac{1}{n}\sum _{i=1} ^{n} \bigl\{ E\bigl[T_{v} \bigl(({\psi }_{j,k})'\bigr) (S_{i}) \bigr]-T_{v} \bigl(({\psi }_{j,k})'\bigr) (S _{i}) \bigr\} =\frac{1}{n}\sum_{i=1}^{n}Y_{i}, $$

where $Y_{i}:=E[T_{v}(({\psi }_{j,k})')(S_{i})]-T_{v} (({\psi }_{j,k})')(S _{i})$.

Similar to (12) and (13),

$$\begin{aligned} \vert Y_{i} \vert \leq C'_{1}2^{(v+\frac{3}{2})j}:=M \quad\text{and}\quad EY_{i}^{2}\leq C'_{2}2^{(2v+2)j}, \end{aligned}$$

(14)

where

$$\begin{aligned} C'_{1}=2(v+2)!\sum _{u=0}^{v} \bigl\Vert \psi ^{(u+1)} \bigr\Vert _{\infty } \quad\text{and} \quad C'_{2}=(v+1)\bigl[(v+2)! \bigr]^{2}C_{\ast }\sum_{u=0}^{v} \bigl\Vert \psi ^{(u+1)} \bigr\Vert _{2}^{2}. \end{aligned}$$

(15)

Then Bernstein’s inequality tells that

$$\begin{aligned} P\bigl\{ \vert \widehat{\beta }_{j,k}- {\beta }_{j,k} \vert >\varUpsilon \lambda _{j}/2\bigr\} =P \Biggl\{ \Biggl\vert \frac{1}{n} \sum_{i=1}^{n}Y_{i} \Biggr\vert >\varUpsilon \lambda _{j}/2\Biggr\} \leq 2\exp \biggl\{ -\frac{n(\varUpsilon \lambda _{j}/2)^{2}}{2(EY_{i} ^{2}+M\varUpsilon \lambda _{j}/6)} \biggr\} . \end{aligned}$$

(16)

On the other hand, combining with (14), $\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}$ and $j2^{j}\leq n$, one shows

$$\begin{aligned} EY_{i}^{2}+M\varUpsilon \lambda _{j}/6 &\leq C'_{2}2^{(2v+2)j}+\frac{C'_{1} \varUpsilon }{6}2^{(v+\frac{3}{2})j} 2^{(v+1)j}\sqrt{\frac{j}{n}} \\ &=2^{(2v+2)j} \biggl(C'_{2}+\frac{C'_{1}\varUpsilon }{6} \sqrt{ \frac{ j2^{j}}{n}} \biggr) \leq \bigl(C'_{2}+C'_{1} \varUpsilon \bigr)2^{(2v+2)j}. \end{aligned}$$

This with (15), $c\geq \max \{8C_{\min },1\}$ implies that

$$\begin{aligned} \frac{n(\varUpsilon \lambda _{j}/2)^{2}}{2(EY_{i}^{2}+M{\varUpsilon \lambda _{j}}/6)} \geq \frac{\varUpsilon ^{2}j}{8(C'_{2}+C'_{1}\varUpsilon )}=\frac{(c \gamma )^{2}j}{8(C'_{2}+C'_{1}c\gamma )} \geq \gamma j\ln 2 \end{aligned}$$

(17)

thanks to $j>0$ and $\gamma >1$.

Hence, it follows from (16)–(17) that

$$\begin{aligned} P\bigl\{ | \widehat{\beta }_{j,k}- {\beta }_{j,k}| > \varUpsilon \lambda _{j}/2\bigr\} \leq 2\exp \biggl\{ -\frac{(c\gamma )^{2}j}{8(C'_{2}+C'_{1}c \gamma )} \biggr\} \lesssim 2^{-\gamma j}, \end{aligned}$$

which is the conclusion of Lemma 4.3. □

At the end of this section, we list two more lemmas which will play key roles in the proof of Theorem 3.3.

Lemma 4.4

([5])

Let $g\in B_{r,q}^{s}(\mathbb{R})$ and $f(x)=g(bx)\ (b\geq 1)$. Then

$$\begin{aligned} \Vert f \Vert _{B_{r,q}^{s}}\leq b^{s-\frac{1}{r}} \Vert g \Vert _{B_{r,q}^{s}}. \end{aligned}$$

To state the last lemma, we need a concept: Let P and Q be two probability measures on ($\varOmega ,\aleph$) and P be absolutely continuous with respect to Q (denoted by $P\ll Q$), the Kullback–Leibler divergence is defined by

$$\begin{aligned} K(P,Q):= \int _{p\cdot q>0} p(x)\ln {\frac{p(x)}{q(x)}}\,dx, \end{aligned}$$

where p and q are density functions of $P,Q$, respectively.

Lemma 4.5

(Fano’s lemma, [10])

Let $( \varOmega , \aleph , P_{k})$ be probability measurable spaces and $A_{k}\in \aleph , k=0,1,\ldots ,m$. If $A_{k}\cap A_{v}=\emptyset$ for $k\neq v$, then

$$\begin{aligned} \sup_{0\leq k\leq m} P_{k}\bigl(A_{k}^{c} \bigr)\geq \min \biggl\{ \frac{1}{2}, \sqrt{m} \exp \bigl(-3e^{-1}- \kappa _{m}\bigr)\biggr\} , \end{aligned}$$

where $A^{c}$ stands for the complement of A and $\kappa _{m}= \inf_{0\leq v\leq m} \frac{1}{m} \sum_{k\neq v}K(P_{k}, P _{v})$.

5 Proofs of results

In this section, we will prove our main results.

5.1 Proofs of upper bounds

We rewrite Theorem 3.1 as follows before giving its proof.

Theorem 5.1

Assume $r\in [1,+\infty ), q\in [1,+\infty ]$ and $s>\frac{1}{r}$, then, for $p\in [1,+\infty )$, the estimator $\widehat{f'_{{\sigma }^{2}}} ^{\mathrm{lin}}$ in (7) with $2^{j_{0}}\sim n^{ \frac{1}{2s'+2(v+1)+1}}$ satisfies

$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}, \end{aligned}$$

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

Proof

When $r>p$, $s'=s-(\frac{1}{r}-\frac{1}{p})_{+}=s$. Denote $\varOmega = \operatorname{supp} (\widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}})$. Then

$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} &=E \int \bigl\vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\vert ^{p}\,dx \\ &\leq E \biggl[ \int _{\varOmega }\bigl( \bigl\vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{ {\sigma }^{2}} \bigr\vert ^{p}\bigr) ^{\frac{r}{p}}\,dx \biggr]^{\frac{p}{r}} \biggl( \int _{\varOmega } 1\,dx\biggr)^{1-\frac{p}{r}}\lesssim E\bigl( \bigl\Vert \widehat{f'_{{\sigma }^{2}}} ^{\mathrm{lin}}-f'_{ {\sigma }^{2}} \bigr\Vert _{r}^{r}\bigr)^{\frac{p}{r}} \end{aligned}$$

due to the Hölder inequality. Furthermore, according to Jensen’s inequality and $\frac{p}{r}<1$,

$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} \bigl(E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{r}^{r}\bigr)^{ \frac{p}{r}}. \end{aligned}$$

(18)

By $s'=s-\frac{1}{r}+\frac{1}{p}\leq s$ and $r\leq p$, one finds $B_{r,q}^{s}\hookrightarrow {B_{p,q}^{s'}}$. Hence,

$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim \sup_{f'_{{\sigma }^{2}}\in B_{p,q}^{s'}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}. \end{aligned}$$

(19)

Next, one only need estimate $\sup_{f'_{{\sigma }^{2}}\in B_{p,q}^{s'}(M)} E\| \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}}\|_{p}^{p}$ by (18) and (19). Note that

$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} & \leq E \bigl[ \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{ {\sigma }^{2}} \bigr\Vert _{p}+ \bigl\Vert P_{j_{0}}f'_{{\sigma }^{2}}-f'_{ {\sigma }^{2}} \bigr\Vert _{p} \bigr]^{p} \\ &\lesssim E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{ {\sigma }^{2}} \bigr\Vert _{p}^{p}+ \bigl\Vert P_{j_{0}}f'_{{\sigma }^{2}}-f'_{ {\sigma }^{2}} \bigr\Vert _{p}^{p}. \end{aligned}$$

(20)

Combining $2^{j_{0}}\sim n^{\frac{1}{2s'+2(v+1)+1}}$, $f'_{{\sigma } ^{2}}\in B_{p,q}^{s'}(M)$ with Lemma 2.1, one concludes

$$\begin{aligned} \bigl\Vert P_{j_{0}}f'_{{\sigma }^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim 2^{-j _{0}s'p}\lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}. \end{aligned}$$

(21)

On the other hand,

$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim 2^{j_{0}(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j_{0}}} E \vert \widehat{\alpha }_{j_{0},k}-{\alpha } _{j_{0},k} \vert ^{p}\lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})j_{0}p} \end{aligned}$$

(22)

thanks to Lemma 4.1 and Lemma 4.2. Then it follows

$$\begin{aligned} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-P_{j_{0}}f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}} \end{aligned}$$

from $2^{j_{0}}\sim n^{\frac{1}{2s'+2(v+1)+1}}$. This with (20) and (21) leads to

$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{p,q}^{s'}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}. \end{aligned}$$

(23)

Combining (23) with (18) and (19), one finds that

$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{lin}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim n^{-\frac{s'p}{2s'+2(v+1)+1}}. \end{aligned}$$

The proof is done. □

Now, the upper bound of nonlinear wavelet estimator (Theorem 3.2) is restated below.

Theorem 5.2

Let $r\in [1,+\infty ), q\in [1,+\infty ]$ and $s>\frac{1}{r}$. Then, for any $p\in [1,+\infty )$, the estimator $\widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}$ in (9) satisfies

$$\begin{aligned} {\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)}} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim (\ln n)^{p} \bigl(n^{-1}\ln n\bigr)^{\alpha p}, \end{aligned}$$

where $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}$.

Proof

When $r>p$, similar to (18),

$$\begin{aligned} \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p}\lesssim \sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} \bigl(E \bigl\Vert \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}} \bigr\Vert _{r}^{r}\bigr)^{ \frac{p}{r}}. \end{aligned}$$

Hence, it suffices to establish the result for $r\leq p$. According to (1), (2) and (9), $E\| \widehat{f'_{{\sigma }^{2}}}^{\mathrm{non}}-f'_{{\sigma }^{2}}\|_{p}^{p}\lesssim A_{1}+A_{2}+A_{3}$, where

$$\begin{array}{rlr}& A_1=E{\parallel \sum _{k\in {\Omega }_{\tau }}({\hat{\alpha }}_{j,k}-{\alpha }_{j,k}){\varphi }_{j,k}\parallel }_p^p;& A_2=E{\parallel \sum _{j=\tau }^{j_1}\sum _{k\in {\Omega }_j}({\tilde{\beta }}_{j,k}-{\beta }_{j,k}){\psi }_{j,k}\parallel }_p^p\text{and}\\ & A_3={\parallel P_{j_1+1}{f}_{{\sigma }^2}^{\prime }-f_{{\sigma }^2}^{\prime }\parallel }_p^p.\end{array}$$

Next, one proves $A_{1}+A_{2}+A_{3}\lesssim (\ln n)^{p}(n^{-1}\ln n)^{ \alpha p}$ for $f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)$ and $r\leq p$.

By the same arguments as (22),

$$\begin{aligned} A_{1}\lesssim 2^{\tau (\frac{p}{2}-1)}\sum_{k\in \varOmega _{ \tau }} E \vert \widehat{\alpha }_{\tau ,k}-{\alpha }_{\tau ,k} \vert ^{p}\lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})\tau p}\thicksim n^{-\frac{p}{2}} \lesssim \bigl(n^{-1}\ln n\bigr)^{\alpha p} \end{aligned}$$

thanks to $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}<\frac{1}{2}$.

Note that $f'_{{\sigma }^{2}}\in B_{r,q}^{s}\hookrightarrow B_{p,q} ^{s-\frac{1}{r}+\frac{1}{p}}$ for $r\leq p$. This with Lemma 2.1 and $2^{j_{1}}\thicksim (\frac{n}{\ln n})^{ \frac{1}{2(v+1)+1}}$ shows

$$\begin{aligned} A_{3}= \bigl\Vert P_{j_{1}+1}f'_{{\sigma }^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \lesssim 2^{-j_{1}(s-\frac{1}{r}+\frac{1}{p})p}\lesssim \biggl( \frac{\ln n}{n}\biggr)^{\frac{(s-\frac{1}{r}+\frac{1}{p})p}{2(v+1)+1}} \lesssim \bigl(n^{-1}\ln n\bigr)^{\alpha p}, \end{aligned}$$

because $s>\frac{1}{r}$ and $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}\leq \frac{s- \frac{1}{r}+\frac{1}{p}}{2(v+1)+1}$.

To estimate $A_{2}$, define

$$\begin{aligned} \widehat{B}_{j}=\bigl\{ k: \vert \widehat{\beta }_{j,k} \vert \geq \varUpsilon \lambda _{j} \bigr\} ;\qquad {B}_{j}= \biggl\{ k: \vert {\beta }_{j,k} \vert \geq \frac{1}{2} \varUpsilon \lambda _{j} \biggr\} \quad\text{and}\quad {C}_{j}=\bigl\{ k: \vert \beta _{j,k} \vert \geq 2\varUpsilon \lambda _{j}\bigr\} . \end{aligned}$$

Then $E\|\sum_{j={\tau }}^{j_{1}} \sum_{k\in \varOmega _{j}}( \widetilde{\beta }_{j,k}-{\beta }_{j,k}) {\psi }_{j,k}\|_{p}^{p} \lesssim (\ln n)^{p-1}\sum_{i=1}^{4}Ee_{i}$ by Lemma 4.1, where

$$\begin{aligned} &e_{1}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B}_{j}\cap {B}_{j}^{c}\bigr\} ; \\ &e_{2}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p} I\{k\in \widehat{B}_{j}\cap {B}_{j}\}; \\ &e_{3}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert {\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}\bigr\} ; \\ &e_{4}={\sum_{j={\tau }}^{j_{1}}} 2^{j(\frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} \vert {\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c}\bigr\} . \end{aligned}$$

By the Hölder inequality and $\{k\in \widehat{B}_{j}\cap {B} _{j}^{c}\}\subseteq \{|\widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}/2\}$,

$$\begin{aligned} E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p}I\bigl\{ k\in \widehat{B}_{j} \cap {B}_{j}^{c} \bigr\} \leq \bigl(E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{2p}\bigr)^{ \frac{1}{2}} P^{\frac{1}{2}} \bigl\{ \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert > \varUpsilon \lambda _{j}/2\bigr\} . \end{aligned}$$

This with Lemma 4.2 and Lemma 4.3 shows that

$$\begin{aligned} Ee_{1}\lesssim \sum_{j={\tau }}^{j_{1}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} n^{-\frac{p}{2}} 2^{j[(v+1)p-\frac{ \gamma }{2}]} \lesssim n^{-\frac{p}{2}}2^{\tau (vp+\frac{3}{2}p-\frac{ \gamma }{2})}\lesssim \biggl(\frac{\ln n}{n} \biggr)^{\alpha p}, \end{aligned}$$

where one uses $\gamma >p(2v+3)$ and $\alpha =\min \{ \frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}\}<\frac{1}{2}$.

From $k\in \widehat{B}_{j}^{c}\cap {C}_{j}$, one finds $| \widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}$ and $|\beta _{j,k}|\leq |\widehat{\beta }_{j,k}- {\beta }_{j,k}|+| \widehat{\beta }_{j,k}|\leq 2|\widehat{\beta }_{j,k}- {\beta }_{j,k}|$. On the other hand, $\{k\in \widehat{B}_{j}^{c}\cap {C}_{j}\}\subseteq \{|\widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}\} \subseteq \{|\widehat{\beta }_{j,k}- {\beta }_{j,k}|>\varUpsilon \lambda _{j}/2\}$. Therefore, it follows from the same arguments as $Ee_{1}$ that

$$\begin{aligned} Ee_{3}\lesssim \sum_{j={\tau }}^{j_{1}} 2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p}I \bigl\{ k\in \widehat{B}_{j}^{c}\cap {C}_{j}\bigr\} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{ \alpha p}. \end{aligned}$$

Next, one estimates $Ee_{2}$ and $Ee_{4}$. Define

$$\begin{aligned} \begin{aligned} &\omega =sr+\biggl(v+\frac{3}{2}\biggr)r-\biggl(v+ \frac{3}{2}\biggr)p,\qquad 2^{j_{0}^{*}}\sim \biggl(\frac{n}{ \ln n} \biggr)^{\frac{1}{2s+2(v+1)+1}},\\ & 2^{j_{1}^{*}}\sim \biggl(\frac{n}{\ln n} \biggr)^{\frac{1}{2(s- \frac{1}{r})+2(v+1)+1}}. \end{aligned} \end{aligned}$$

(24)

Then, by $s>\frac{1}{r}$ and $2^{j_{1}}\thicksim (\frac{n}{\ln n})^{ \frac{1}{2(v+1)+1}}$,

$$\begin{aligned} 0< \frac{1}{2s+2(v+1)+1},~\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}< \frac{1}{2(v+1)+1} \quad\text{and}\quad \tau < j_{0}^{*}, j_{1}^{*}< j_{1}. \end{aligned}$$

When $\omega \geq 0$, one writes down

$$\begin{aligned} e_{2}=\Biggl(\sum_{j={\tau }}^{j_{0}^{*}}+ \sum_{j=j_{0}^{*}} ^{j_{1}}\Biggr) 2^{j(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j}} \vert \widehat{\beta }_{j,k}-{ \beta }_{j,k} \vert ^{p}I\{k\in \widehat{B}_{j} \cap {B}_{j}\} :=e_{21}+e_{22}. \end{aligned}$$

(25)

According to (22),

$$\begin{aligned} Ee_{21}\lesssim \sum_{j={\tau }}^{j_{0}^{*}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p} \lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})j_{0}^{*}p}. \end{aligned}$$

(26)

Note that $\frac{2|\beta _{jk}|}{\varUpsilon \lambda _{j}}\geq 1$, $\sum_{k}|\beta _{jk}|^{r}\lesssim 2^{-j(s+\frac{1}{2}-\frac{1}{r})r}$ from $k\in B_{j}$, $f'_{\sigma ^{2}}\in B_{r,q}^{s}(M)$ and Lemma 2.1; On the other hand, Lemma 4.2 tells $E|\widehat{\beta }_{j,k}-{\beta }_{j,k}|^{p}\lesssim 2^{(v+1)jp} n ^{-\frac{p}{2}}$. These with $\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}$ and $\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p\geq 0$ lead to

$$\begin{aligned} Ee_{22} &\lesssim \sum_{j=j_{0}^{*}}^{j_{1}} 2^{j( \frac{p}{2}-1)}\sum_{k\in \varOmega _{j}} E \vert \widehat{\beta }_{j,k}- {\beta }_{j,k} \vert ^{p}\biggl( \frac{ \vert \beta _{jk} \vert }{\varUpsilon \lambda _{j}}\biggr)^{r} \\ &\lesssim \sum_{j=j_{0}^{*}}^{j_{1}}2^{j(\frac{p}{2}-1)}2^{(v+1)jp} n^{-\frac{p}{2}}\lambda _{j}^{-r}\sum _{k} \vert \beta _{jk} \vert ^{r} \lesssim 2^{-j _{0}^{*}\omega }n^{-\frac{p-r}{2}}. \end{aligned}$$

(27)

Combining (25)–(27) with $2^{j_{0}^{*}}\sim (\frac{n}{ \ln n})^{\frac{1}{2s+2(v+1)+1}}$, one obtains

$$\begin{aligned} Ee_{2}=Ee_{21}+Ee_{22}\lesssim n^{-\frac{p}{2}}2^{(v+1+\frac{1}{2})j _{0}^{*}p}+2^{-j_{0}^{*}\omega }n^{-\frac{p-r}{2}} \lesssim \biggl(\frac{ \ln n}{n}\biggr)^{\frac{sp}{2s+2(v+1)+1}}=\biggl(\frac{\ln n}{n} \biggr)^{\alpha p} \end{aligned}$$

because of $\omega \geq 0$ and $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}= \frac{s}{2s+2(v+1)+1}$.

When $\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p<0$, $\alpha = \min \{\frac{s}{2s+2(v+1)+1}, \frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}\}=\frac{s-\frac{1}{r}+ \frac{1}{p}}{2(s- \frac{1}{r})+2(v+1)+1}$. Define $p_{1}=(1-2\alpha )p$. Then $r\leq p_{1}\leq p$ follows from

$$\begin{aligned} \omega < 0 \quad\text{and}\quad r\leq p_{1}=(1-2\alpha )p= \frac{2(v+1)p+p-2}{2(s-\frac{1}{r})+2(v+1)+1}\leq p. \end{aligned}$$

Moreover, $\sum_{k}|\beta _{jk}|^{p_{1}}\leq (\sum_{k}|\beta _{jk}|^{r})^{\frac{p _{1}}{r}} \lesssim 2^{-j(s+\frac{1}{2}-\frac{1}{r})p_{1}}$ thanks to $r\leq p_{1}$, $f'_{\sigma ^{2}}\in B_{r,q}^{s}(M)$ and Lemma 2.1. This with (27) and $\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}$ shows

$$\begin{aligned} Ee_{2} &\lesssim \sum_{j=\tau }^{j_{1}} 2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} E \vert \widehat{\beta }_{j,k}-{\beta }_{j,k} \vert ^{p}\biggl( \frac{ \vert \beta _{jk} \vert }{\varUpsilon \lambda _{j}}\biggr)^{p_{1}} \\ &\lesssim \sum_{j=\tau }^{j_{1}}2^{j(\frac{p}{2}-1)}2^{(v+1)jp} n^{-\frac{p}{2}}\lambda _{j}^{-p_{1}}\sum _{k} \vert \beta _{jk} \vert ^{p_{1}} \\ &\lesssim n^{-\frac{p-p_{1}}{2}}\sum_{j=\tau }^{{j_{1}}} 2^{j[ \frac{p}{2}-1+(v+1)(p-p_{1})-(s+\frac{1}{2}-\frac{1}{r})p_{1}]} \lesssim \ln n \biggl(\frac{\ln n}{n}\biggr)^{\alpha p} \end{aligned}$$

due to $\frac{p-p_{1}}{2}=\alpha p$ and $\frac{p}{2}-1+(v+1)(p-p_{1})-(s+ \frac{1}{2}-\frac{1}{r})p_{1}=0$.

Finally, one estimates $Ee_{4}$. When $\omega =sr+(v+\frac{3}{2})r-(v+ \frac{3}{2})p\geq 0$, $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}= \frac{s}{2s+2(v+1)+1}$. Furthermore,

$$\begin{aligned} e_{4}=\Biggl(\sum_{j={\tau }}^{j_{0}^{*}}+ \sum_{j={j_{0}^{*}}+1} ^{j_{1}}\Biggr) 2^{j(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j}} \vert {\beta } _{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B}_{j}^{c}\cap {C}_{j}^{c}\bigr\} :=e_{41}+e _{42}, \end{aligned}$$

(28)

where $j_{0}^{*}$ is given by (24).

Since $|\beta _{j,k}|\leq 2\varUpsilon \lambda _{j}\lesssim 2^{(v+1)j}(jn ^{-1})^{\frac{1}{2}}$ holds by $k\in C_{j}^{c}$ and $\lambda _{j}=2^{(v+1)j}\sqrt{ \frac{j}{n}}$, one concludes that

$$\begin{aligned} Ee_{41} &= E\sum_{j={\tau }}^{j_{0}^{*}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p}I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c}\bigr\} \\ &\lesssim \sum_{j={\tau }}^{j_{0}^{*}}2^{j(\frac{p}{2}-1)}2^{j} 2^{(v+1)jp}\bigl(jn^{-1}\bigr)^{\frac{p}{2}}\lesssim 2^{(v+1+\frac{1}{2})j_{0} ^{*}p}\biggl(\frac{\ln n}{n}\biggr)^{\frac{p}{2}}. \end{aligned}$$

(29)

On the other hand,

$$\begin{aligned} Ee_{42} &= E\sum_{j={j_{0}^{*}+1}}^{j_{1}} 2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c}\bigr\} \\ &\lesssim \sum_{j=j_{0}^{*}+1}^{j_{1}}2^{j(\frac{p}{2}-1)} \sum_{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p}\bigl(\lambda _{j} \vert \beta _{j,k} \vert ^{-1}\bigr)^{p-r} \\ &\lesssim \sum_{j={j_{0}^{*}+1}}^{j_{1}}2^{j(\frac{p}{2}-1)} \lambda _{j}^{p-r}\sum_{k} \vert \beta _{j,k} \vert ^{r} \end{aligned}$$

due to $|\beta _{j,k}|\leq 2\varUpsilon \lambda _{j}$ and $r\leq p$.

Clearly, $\|{\beta }_{j,\cdot }\|_{l_{r}} \lesssim 2^{-j(s-\frac{1}{r}+ \frac{1}{2})}$ by $f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)$ and Lemma 2.1. This with $\lambda _{j}=2^{(v+1)j}\sqrt{\frac{j}{n}}$ implies that

$$\begin{aligned} Ee_{42} \lesssim \sum_{j={j_{0}^{*}+1}}^{j_{1}}2^{j( \frac{p}{2}-1)} 2^{(v+1)(p-r)j}2^{-j(sr+\frac{r}{2}-1)}\biggl( \frac{\ln n}{n}\biggr)^{\frac{p-r}{2}} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{ \frac{p-r}{2}}2^{-j_{0}^{\ast }\omega }, \end{aligned}$$

(30)

because $\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p\geq 0$.

According to (28)–(30) and $2^{j_{0}^{*}}\sim (\frac{n}{ \ln n})^{\frac{1}{2s+2(v+1)+1}}$, one obtains

$$\begin{aligned} Ee_{4}=Ee_{41}+Ee_{42}\lesssim 2^{(v+1+\frac{1}{2})j_{0}^{*}p} \biggl(\frac{ \ln n}{n}\biggr)^{\frac{p}{2}}+\biggl(\frac{\ln n}{n} \biggr)^{\frac{p-r}{2}}2^{-j_{0} ^{\ast }\omega } \lesssim \biggl(\frac{\ln n}{n} \biggr)^{\alpha p} \end{aligned}$$

by $\alpha =\frac{s}{2s+2(v+1)+1}$.

For the case of $\omega =sr+(v+\frac{3}{2})r-(v+\frac{3}{2})p<0$. Let

$$\begin{aligned} e_{4}=\Biggl(\sum_{j={\tau }}^{j_{1}^{*}}+ \sum_{j={j_{1}^{*}}+1} ^{j_{1}}\Biggr) 2^{j(\frac{p}{2}-1)}\sum _{k\in \varOmega _{j}} \vert {\beta } _{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B}_{j}^{c}\cap {C}_{j}^{c}\bigr\} :=e_{41}'+e _{42}', \end{aligned}$$

(31)

where $j_{1}^{*}$ is given by (24). Similar to (30),

$$\begin{aligned} Ee_{41}'=\sum _{j={\tau }}^{j_{1}^{*}} 2^{j(\frac{p}{2}-1)} \sum _{k\in \varOmega _{j}} \vert {\beta }_{j,k} \vert ^{p} I\bigl\{ k\in \widehat{B} _{j}^{c}\cap {C}_{j}^{c} \bigr\} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{\frac{p-r}{2}}2^{-j _{1}^{\ast }\omega } \end{aligned}$$

(32)

thanks to $\omega <0$.

To estimate $Ee_{42}'$, one observes that $\|\beta _{j,\cdot }\|_{l_{p}} \lesssim \|\beta _{j,\cdot }\|_{l_{r}}\lesssim 2^{-j(s-\frac{1}{r}+ \frac{1}{2})}$ by $r\leq p$, $f'_{\sigma ^{2}}\in B_{r,q}^{s}(M)$ and Lemma 2.1. Hence,

$$\begin{aligned} Ee_{42}' &=\sum _{j={j_{1}^{*}+1}}^{j_{1}}2^{j(\frac{p}{2}-1)} \sum _{k\in \varOmega _{j}} \vert \beta _{j,k} \vert ^{p}I \bigl\{ k\in \widehat{B}_{j} ^{c}\cap {C}_{j}^{c} \bigr\} \lesssim \sum_{j={j_{1}^{*}+1}}^{j_{1}}2^{j( \frac{p}{2}-1)} \Vert \beta _{j,\cdot } \Vert _{l_{r}}^{p} \\ &\lesssim \sum_{j={j_{1}^{*}+1}}^{j_{1}}2^{j(\frac{p}{2}-1)}2^{-j(s- \frac{1}{r}+\frac{1}{2})p} \lesssim 2^{-j_{1}^{*}(s-\frac{1}{r}+ \frac{1}{p})p} \end{aligned}$$

(33)

because of $s>\frac{1}{r}$.

Combining (31)–(33) with $2^{j_{1}^{*}}\sim (\frac{n}{ \ln n})^{\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}}$, one knows

$$\begin{aligned} Ee_{4}=Ee_{41}'+Ee_{42}' \lesssim \biggl(\frac{\ln n}{n}\biggr)^{\frac{p-r}{2}}2^{-j _{1}^{\ast }\omega } +2^{-j_{1}^{*}(s-\frac{1}{r}+\frac{1}{p})p} \lesssim \biggl(\frac{\ln n}{n}\biggr)^{\alpha p} \end{aligned}$$

thanks to $\omega <0$ and $\alpha =\min \{\frac{s}{2s+2(v+1)+1}, \frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}\}=\frac{s- \frac{1}{r}+ \frac{1}{p}}{2(s-\frac{1}{r})+2(v+1)+1}$. This completes the proof of Theorem 5.2. □

5.2 Proof of lower bound

Finally, we are in a position to state and prove the lower bound estimation.

Theorem 5.3

Assume $s>0$ and $r,q\in [1,+\infty ]$, then, for any $p\in [1,+ \infty )$,

$$\begin{aligned} \inf_{\widehat{f}'_{{\sigma }^{2}}}\sup_{f'_{{\sigma }^{2}}\in B_{r,q}^{s}(M)} E \bigl\Vert \widehat{f}'_{{\sigma } ^{2}}-f'_{{\sigma }^{2}} \bigr\Vert _{p}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r}+ \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}, \end{aligned}$$

where $\widehat{f}'_{{\sigma }^{2}}$ runs over all possible estimators of $f'_{{\sigma }^{2}}$.

Proof

It is sufficient to construct density functions $h_{k}$ such that $h'_{k}\in B_{r,q}^{s}(M)$ and

$$\begin{aligned} \sup_{k} E \bigl\Vert \widehat{f}'_{{\sigma }^{2}}-h'_{k} \bigr\Vert _{p}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r}+ \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}. \end{aligned}$$

Define $g(x)=Cm(x)$, where $m\in C_{0}^{\infty }$ with $\operatorname{supp} m \subseteq [0,1]$, $\int _{\mathbb{R}} m(x)\,dx=0$ and $C>0$ is a constant. Let $C_{0}^{\infty }$ stand for the set of all infinitely many times differentiable and compactly supported functions. Furthermore, one chooses a density function $h_{0}$ satisfying $h_{0}\in B_{r,q}^{s+1}( \frac{M}{2}), \operatorname{supp} h_{0}\subseteq [0,1]$ and $h_{0}(x)\geq M _{1}>0$ for $x\in [\frac{1}{2},\frac{3}{4}]$.

Take $a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}$ and

$$\begin{aligned} h_{1}(x)=h_{0}(x)+a_{j}G_{v}(g_{j,l}) (x), \end{aligned}$$

(34)

where G is given by (5) and $g_{j,l}(x)=2^{\frac{j}{2}}g(2^{j}x-l)$ with $l=2^{j-1}$.

First, one checks that $h_{1}$ is a density function. Since $\operatorname{supp} g_{j,l}\subseteq [\frac{1}{2},\frac{3}{4}]$ by $\operatorname{supp} m\subseteq [0,1]$ and j large enough, one finds $h_{1}(x)\geq 0$ for $x\notin [\frac{1}{2},\frac{3}{4}]$. It is easy to calculate that

$$\begin{aligned} G_{v}(g_{j,l}) (x)=(-1)^{v}\sum _{u=1}^{v}C_{u}x^{u}(g_{j,l})^{(u)}(x), \end{aligned}$$

(35)

where $C_{u}>0$ is a constant. Then, for $x\in [\frac{1}{2}, \frac{3}{4}]$ and large j,

$$\begin{aligned} h_{1}(x) &\geq M_{1}- \Biggl\vert a_{j}\sum_{u=1}^{v}C_{u}x^{u}(g_{j,l})^{(u)}(x) \Biggr\vert \\ &\geq M_{1}-a_{j}2^{\frac{j}{2}}\sum _{u=1}^{v}C_{u}2^{uj} \bigl\Vert {g}^{(u)}\bigl(2^{j}\cdot -l\bigr) \bigr\Vert _{\infty } \\ &\geq M_{1}-2^{-j(s-\frac{1}{r}+1)}\sum_{u=1}^{v}C_{u} \bigl\Vert g ^{(u)} \bigr\Vert _{\infty }\geq 0 \end{aligned}$$

(36)

thanks to $h_{0}(x)|_{[\frac{1}{2},\frac{3}{4}]}\geq M_{1}$ and $a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}$. On the other hand, $\int _{\mathbb{R}} g(x)\,dx=\int _{\mathbb{R}} Cm(x)\,dx=0$ and $\operatorname{supp} g_{j,l}\subseteq [\frac{1}{2},\frac{1}{2}+2^{-j}]$ by $\operatorname{supp} m\subseteq [0,1]$ and $l=2^{j-1}$. This with $m\in C_{0}^{\infty }$ and $g(x)=Cm(x)$ shows

$$\begin{aligned} \int x^{u}(g_{j,l})^{(u)}(x) \,dx&=x^{u}(g_{j,l})^{(u-1)} (x) | _{\frac{1}{2}}^{\frac{1}{2}+2^{-j}}-u \int x^{u-1}(g_{j,l})^{(u-1)}(x)\,dx \\ &=\cdots =(-1)^{m}\frac{u!}{(u-m)!} \int x^{u-m}(g_{j,l})^{(u-m)} (x) \,dx\\ &=(-1)^{u}u! \int g_{j,l}(x)\,dx=0 \end{aligned}$$

for any $u\in \{1,\ldots , v\}$. Therefore,

$$\begin{aligned} \int h_{1}(x)\,dx= \int h_{0}(x)\,dx+(-1)^{v}a_{j}\sum _{u=1}^{v}C _{u} \int x^{u}(g_{j,l})^{(u)}(x)\,dx=1. \end{aligned}$$

From this with (36) one concludes that $h_{1}$ is a density function.

Next, one shows $h'_{0}, h'_{1}\in B_{r,q}^{s}(M)$. Clearly, $h'_{0}\in B_{r,q}^{s}(M)$ by $h_{0}\in B_{r,q}^{s+1}(\frac{M}{2})$. Hence, one only needs prove $h'_{1}\in B_{r,q}^{s}(M)$.

By (34) and (35),

$$\begin{aligned} \Vert h_{1} \Vert _{B_{r,q}^{s+1}}\leq \Vert h_{0} \Vert _{B_{r,q}^{s+1}}+a_{j}\sum _{u=1}^{v}C_{u} \bigl\Vert x^{u}(g_{j,l})^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}}. \end{aligned}$$

(37)

On the other hand, for each $\tau \in \{0,\ldots ,u\}$,

$$\begin{aligned} \biggl\Vert \biggl[2^{j}\biggl(x-\frac{1}{2}\biggr) \biggr]^{u-\tau }g^{(u)}\biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{B_{r,q}^{s+1}} \leq 2^{j(s+1-\frac{1}{r})} \bigl\Vert x^{u-\tau }g^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}} \end{aligned}$$

because of Lemma 4.4. Combining this with $l=2^{j-1}$ and $[2^{j}(x-\frac{1}{2}+\frac{1}{2})]^{u}= \sum_{\tau =0}^{u}C ^{\tau }_{u}[2^{j}(x-\frac{1}{2})]^{u-\tau }2^{-\tau }2^{\tau j}$, one obtains

$$\begin{aligned} \bigl\Vert x^{u}(g_{j,l})^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}} &=2^{\frac{j}{2}} \biggl\Vert \biggl[2^{j}\biggl(x- \frac{1}{2}+\frac{1}{2}\biggr) \biggr]^{u}g^{(u)}\biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{B_{r,q} ^{s+1}} \\ &\leq 2^{(u+\frac{1}{2})j}\sum_{\tau =0}^{u}C^{\tau }_{u} \biggl\Vert \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr) \biggr]^{u-\tau }g^{(u)}\biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{B_{r,q}^{s+1}} \\ &\leq 2^{j(s-\frac{1}{r}+\frac{1}{2}+u+1)}\sum_{\tau =0}^{u}C ^{\tau }_{u} \bigl\Vert x^{u-\tau }g^{(u)}(x) \bigr\Vert _{B_{r,q}^{s+1}}. \end{aligned}$$

(38)

Denote $M'=\max \{C_{u},C_{u}^{\tau }: \tau =0,\ldots ,u; u=1,\ldots ,v\}$. Then there exists a constant $C>0$ such that

$$\begin{aligned} x^{u-\tau }g^{(u)}\in B_{r,q}^{s+1}\biggl( \frac{M}{2v^{2}M^{\prime 2}}\biggr) \end{aligned}$$

thanks to $g(x)=Cm(x)$ and $m\in C_{0}^{\infty }\subseteq B_{r,q}^{s+1}$. This with (37) and (38) leads to

$$\begin{aligned} \Vert h_{1} \Vert _{B_{r,q}^{s+1}}\leq \Vert h_{0} \Vert _{B_{r,q}^{s+1}}+a_{j}\sum_{u=1}^{v} uM^{\prime 2}2^{j(s-\frac{1}{r}+\frac{1}{2}+u+1)}\frac{M}{2v ^{2}M^{\prime 2}}\leq \frac{M}{2}+ \frac{M}{2}=M, \end{aligned}$$

because $h_{0}\in B_{r,q}^{s+1}(\frac{M}{2})$ and $a_{j}=2^{-j(s- \frac{1}{r}+\frac{1}{2}+v+1)}$. Therefore, $h_{1}\in B_{r,q}^{s+1}(M)$ and $h'_{1}\in B_{r,q}^{s}(M)$.

According to (35),

$$\begin{aligned} \bigl[G_{v}(g_{j,l})(x)\bigr] '=(-1)^{v} \sum_{u=0}^{v}C'_{u}x^{u}(g_{j,l})^{(u+1)}(x), \end{aligned}$$

where $C'_{u}>0$ is a constant and $l=2^{j-1}$. Hence,

$$\begin{aligned} \bigl\Vert h'_{1}-h'_{0} \bigr\Vert _{p} &= a_{j} \bigl\Vert \bigl[G_{v}(g_{j,l})\bigr]' \bigr\Vert _{p} \\ &= a_{j}2^{\frac{3j}{2}} \Biggl\Vert \sum _{u=0}^{v}C'_{u} \biggl[2^{j}\biggl(x- \frac{1}{2}+\frac{1}{2}\biggr) \biggr]^{u}g^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p}. \end{aligned}$$

(39)

On the other hand, by using $[2^{j}(x-\frac{1}{2}+\frac{1}{2})]^{u}= \sum_{\tau =0}^{u} C^{\tau }_{u}[2^{j}(x-\frac{1}{2})]^{u- \tau }2^{-\tau }2^{\tau j}$, one concludes that

$$\begin{aligned} &\Biggl\Vert \sum_{u=0}^{v}C'_{u} \biggl[2^{j}\biggl(x-\frac{1}{2}+\frac{1}{2}\biggr) \biggr]^{u}g ^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \\ &\quad\geq \Biggl\Vert \sum_{u=0}^{v}C'_{u} 2^{-u}2^{uj}g^{(u+1)}\biggl[2^{j} \biggl(x-\frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \\ &\qquad{}- \Biggl\Vert \sum_{u=0}^{v}C'_{u} \Biggl\{ \sum_{\tau =0}^{u-1}C^{\tau } _{u}\biggl[2^{j} \biggl(x-\frac{1}{2}\biggr) \biggr]^{u-\tau }2^{-\tau } 2^{\tau j}\Biggr\} g^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \end{aligned}$$

(40)

and

$$\begin{aligned} &\Biggl\Vert \sum_{u=0}^{v}C'_{u}2^{-u}2^{uj}g^{(u+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p} \\ &\quad \geq \biggl\Vert C'_{v}2^{-v}2^{vj}g^{(v+1)} \biggl[2^{j}\biggl(x- \frac{1}{2}\biggr)\biggr] \biggr\Vert _{p} \\ &\qquad{}- \Biggl\Vert \sum_{u=0}^{v-1}C'_{u} 2^{-u}2^{uj}g^{(u+1)}\biggl[2^{j} \biggl(x- \frac{1}{2}\biggr)\biggr] \Biggr\Vert _{p}. \end{aligned}$$

(41)

Let $x'=2^{j}(x-\frac{1}{2})$. Then there exists a constant $C'>0$ such that

$$\begin{aligned} \bigl\Vert h'_{1}-h'_{0} \bigr\Vert _{p} \geq {}& a_{j}2^{\frac{3}{2}j}2^{-\frac{j}{p}} \Biggl\{ C'_{v}2^{-v}2^{vj} \bigl\Vert g^{(v+1)} \bigr\Vert _{p}-2^{(v-1)j}\sum _{u=0} ^{v-1}C'_{u}2^{-u} \bigl\Vert g^{(u+1)} \bigr\Vert _{p} \\ &{}-2^{(v-1)j}\sum_{u=0}^{v}C'_{u} \sum_{\tau =0}^{u-1}C ^{\tau }_{u}2^{-\tau } \bigl\Vert \bigl(x'\bigr)^{u-\tau }g^{(u+1)} \bigr\Vert _{p} \Biggr\} \\ \geq {}& C'a_{j}2^{\frac{3}{2}j}2^{-\frac{j}{p}}2^{jv} =C'2^{-j(s- \frac{1}{r}+\frac{1}{p})}:=\delta _{j} \end{aligned}$$

thanks to (39)–(41), $g\in C_{0}^{\infty }$ and $a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}$.

Define $A_{k}=\{ \Vert \widehat{f}'_{{\sigma }^{2}}-h'_{k} \Vert _{p}<\frac{\delta _{j}}{2}\}\ (k\in \{0,1\})$. Then $A_{0}\cap A_{1}= \emptyset$. According to Lemma 4.5,

$$\begin{aligned} \sup_{k\in \{0,1\}} P_{f_{s_{k}}}^{n} \bigl(A_{k}^{c}\bigr)\geq \min \biggl\{ \frac{1}{2}, \exp \bigl(-3e^{-1}-\kappa _{1}\bigr)\biggr\} , \end{aligned}$$

(42)

where $P_{f}^{n}$ stands for the probability measure corresponding to the density function $f^{n}(x):=f(x_{1})f(x_{2})\cdots f(x_{n})$. Hence,

$$\begin{aligned} E \bigl\Vert \widehat{f}'_{\sigma ^{2}}-h'_{k} \bigr\Vert _{p}^{p} \geq \biggl( \frac{\delta _{j}}{2} \biggr)^{p}P_{f_{s_{k}}}^{n}\biggl( \bigl\Vert \widehat{f}'_{{\sigma } ^{2}}-h'_{k} \bigr\Vert _{p}\geq \frac{\delta _{j}}{2}\biggr) =\biggl(\frac{\delta _{j}}{2} \biggr)^{p}P _{f_{s_{k}}}^{n}\bigl(A_{k}^{c} \bigr). \end{aligned}$$

This with (42) implies

$$\begin{aligned} \sup_{k\in \{0,1\}} E \bigl\Vert \widehat{f}'_{\sigma ^{2}}-h'_{k} \bigr\Vert _{p} ^{p}\geq \sup_{k\in \{0,1\}}\biggl( \frac{\delta _{j}}{2}\biggr)^{p}P_{f_{s _{k}}}^{n} \bigl(A_{k}^{c}\bigr)\geq \biggl(\frac{\delta _{j}}{2} \biggr)^{p}\min \biggl\{ \frac{1}{2},\exp \bigl(-3e^{-1}- \kappa _{1}\bigr)\biggr\} . \end{aligned}$$

(43)

Next, one shows $\kappa _{1}\leq C_{0}na_{j}^{2}$. Recall that

$${\kappa }_1=\underset{0\le u\le 1}{inf}\sum _{k\ne u}K(P_{f_{s_k}}^n,P_{f_{s_u}}^n)\le K(P_{f_{s_1}}^n,P_{f_{s_0}}^n).$$

(44)

Then

$$\begin{aligned} K\bigl(P_{f_{s_{1}}}^{n}, P_{f_{s_{0}}}^{n}\bigr) \leq \sum_{i=1}^{n} \int f_{s_{1}}(x_{i})\ln \frac{f_{s_{1}}(x_{i})}{f_{s_{0}}(x_{i})}\,dx _{i}=n \int f_{s_{1}}(x)\ln \frac{f_{s_{1}}(x)}{f_{s_{0}}(x)}\,dx \end{aligned}$$

due to $f_{s_{0}}^{n}(x)=\prod_{j=1}^{n}f_{s_{0}}(x_{j})$ and $f_{s_{1}}^{n}(x)=\prod_{j=1}^{n}f_{s_{1}}(x_{j})$. Since $\ln u \leq u-1$ holds for $u>0$, one knows

$$\begin{aligned} K\bigl(P_{f_{s_{1}}}^{n}, P_{f_{s_{0}}}^{n}\bigr) & \leq n \int f_{s_{1}}(x) \biggl( \frac{f _{s_{1}}(x)}{f_{s_{0}}(x)}-1\biggr)\,dx \\ &= n \int f^{-1}_{s_{0}}(x)\bigl| f_{s_{1}}(x)-f_{s_{0}}(x) \bigr| ^{2} \,dx. \end{aligned}$$

(45)

According to Chesneau’s work in Ref. [8], $f_{s_{k}}(x)=\frac{1}{(v-1)!}\int _{x}^{1}(\ln y-\ln x)^{v-1}h_{k}(y) \frac{1}{y}\,dy$. Then

$$\begin{aligned} f_{s_{1}}(x)-f_{s_{0}}(x) &=\frac{a_{j}}{(v-1)!} \int _{x}^{\frac{1}{2}+2^{-j}}( \ln y-\ln x)^{v-1}G_{v}(g_{j,l}) (y)\frac{1}{y}\,dy \\ &=-\frac{a_{j}}{(v-1)!} \int _{x}^{\frac{1}{2}+2^{-j}}(\ln y-\ln x)^{v-1}\bigl[G _{v-1}(g_{j,l}) (y)\bigr]'\,dy \end{aligned}$$

because of (34) and $G_{v}(g_{j,l})(x)=-x[G_{v-1}(g_{j,l})(x)]'$.

By the formula of integration by parts,

$$\begin{aligned} f_{s_{1}}(x)-f_{s_{0}}(x) &=-\frac{a_{j}}{(v-2)!} \int _{x}^{ \frac{1}{2}+2^{-j}} (\ln y-\ln x)^{v-2} \bigl[G_{v-2}(g_{j,l}) (y)\bigr]'\,dy= \cdots \\ &=-\frac{a_{j}}{(v-m)!} \int _{x}^{\frac{1}{2}+2^{-j}}(\ln y-\ln x)^{v-m} \bigl[G_{v-m}(g_{j,l}) (y)\bigr]'\,dy=\cdots \\ &=-a_{j} \int _{x}^{\frac{1}{2}+2^{-j}}(g_{j,l})'(y) \,dy=a_{j}g_{j,l}(x), \end{aligned}$$

(46)

because $l=2^{j-1}$ and $(\ln y-\ln x)^{v-m}G_{v-m}(g_{j,l})(y)| _{x}^{\frac{1}{2}+2^{-j}}=0$ for any $m\in \{1,\ldots ,v-1\}$. On the other hand, for each $x\in [\frac{1}{2}, \frac{1}{2}+2^{-j}]$ and large j,

$$\begin{aligned} f_{s_{0}}(x) &\geq \frac{M_{1}}{(v-1)!} \int _{x}^{\frac{3}{4}}(\ln y- \ln x)^{v-1} \frac{1}{y}\,dy=\frac{M_{1}}{v!}\biggl(\ln \frac{3}{4}-\ln x \biggr)^{v} \\ &\geq \frac{M_{1}}{v!}\biggl[\ln \frac{3}{4}-\ln \biggl( \frac{1}{2}+2^{-j}\biggr)\biggr]^{v} \geq M_{2}>0 \end{aligned}$$

(47)

thanks to $f_{s_{0}}(x)=\frac{1}{(v-1)!}\int _{x}^{1}(\ln y-\ln x)^{v-1}h _{0}(y) \frac{1}{y}\,dy$ and $h_{0}(x)|_{[\frac{1}{2},\frac{3}{4}]} \geq M_{1}$. Combining with (44)–(47), one obtains

$$\begin{aligned} \kappa _{1}\leq M_{2}^{-1}n \int\bigl| a_{j}g_{j,l}(x)\bigr| ^{2} \,dx \leq C _{0}na_{j}^{2}, \end{aligned}$$

where $C_{0}>0$ is a constant.

Choose $2^{j}\thicksim n^{\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}}$ and recall $a_{j}=2^{-j(s-\frac{1}{r}+\frac{1}{2}+v+1)}$. Then

$$\begin{aligned} \kappa _{1}\lesssim na_{j}^{2}=n2^{-j[2(s-\frac{1}{r})+2(v+1)+1]} \thicksim 1 \quad \text{and} \quad e^{-\kappa _{1}}\gtrsim 1. \end{aligned}$$

Substituting $\delta _{j}\thicksim 2^{-j(s-\frac{1}{r}+\frac{1}{p})}$, $2^{j}\thicksim n^{\frac{1}{2(s-\frac{1}{r})+2(v+1)+1}}$ into (43), one obtains

$$\begin{aligned} \sup_{k\in \{0,1\}} E \bigl\Vert \widehat{f}'_{\sigma ^{2}}-h'_{k} \bigr\Vert _{p} ^{p}\gtrsim \delta _{j}^{p} \gtrsim n^{-\frac{(s-\frac{1}{r} + \frac{1}{p})p}{2(s-\frac{1}{r})+2(v+1)+1}}, \end{aligned}$$

which is the desired conclusion. □