The Berry–Esseen type bounds of the weighted estimator in a nonparametric model with linear process errors

Abstract

In this paper, the Berry–Esseen type bounds of the weighted estimator in a nonparametric regression model are investigated under some mild conditions when random errors are from a linear process generated by $\varphi$-mixing random variables. In particular, the rate of uniform normal approximation is near to $O(n^{-\frac{3}{16}})$ by the choice of some constants, which generalizes and improves the corresponding results of Li et al. (Stat Probab Lett 81:103–110, 2011) and Ding et al. (J Inequal Appl 2018:10, 2018). Finally, the simulation study is provided to verify the validity of the theoretical results.

Appendix

Lemma A.1

(cf. Liang and Fan 2009, Lemma 3.1) Let X and $Y_1,Y_2,\ldots ,Y_m$ be random variables. Then for positive numbers $a_1,a_2,\ldots ,a_m$, we have

$$\begin{aligned} \sup _{u}\left| P\left( X+\sum _{i=1}^mY_i\le u\right) -\Phi (u)\right| \le \sup _{u}\left| P(X\le u)-\Phi (u)\right| +\sum _{i=1}^{m}\frac{a_i}{\sqrt{2\pi }}+\sum _{i=1}^{m}P(|Y_i|>a_i). \end{aligned}$$

Lemma A.2

(cf. Lu and Lin 1997, Lemma 1.2.8) Let $\{X_n,n\ge 1\}$ be a sequence of $\varphi$-mixing random variables. Let $X\in L_p({\mathcal {F}}_1^{k})$, $Y\in L_q({\mathcal {F}}_{k+n}^{\infty })$, $p\ge 1$, $q\ge 1$ and $1/p+1/q=1$. Then

$$\begin{aligned} |EXY-EXEY|\le 2(\varphi (n))^{1/p}(E|X|^p)^{1/p}(E|Y|^q)^{1/q}. \end{aligned}$$

Lemma A.3

(cf. Petrov 1995, Theorem 5.7) Let $X_1,X_2,\ldots ,X_n$ be independent random variables with $EX_j=0$ and $E|X_j|^{2+\delta }<\infty$ for some $0<\delta \le 1$ and $j=1,2,\ldots ,n$. Denote $B_n=\sum \nolimits _{j=1}^n \text {Var} X_j$, then

$$\begin{aligned} \sup _{x}\left| P\left( \sum _{j=1}^nX_j\le x\right) -\Phi (x)\right| \le CB_n^{-1-\delta /2}\sum _{j=1}^n E|X_j|^{2+\delta }. \end{aligned}$$

Lemma A.4

(cf. Yang 1995, Lemma 2) Let $\{X_n,n\ge 1\}$ be a sequence of $\varphi$-mixing random variables with $EX_j=0$. If $\sum \nolimits _{j=1}^{\infty }\varphi ^{1/2}(j)<\infty$ and $E|X_j|^q<\infty$ for $q\ge 2$ and $j=1,2,\ldots$, then for each $n\ge 1$,

$$\begin{aligned} E\left| \sum _{j=1}^n X_j\right| ^q\le \left\{ \sum _{j=1}^nE|X_j|^q+\left( \sum _{j=1}^nEX_j^2\right) ^{q/2}\right\} \end{aligned}$$

Lemma A.5

(cf. Li et al. 2008, Lemma 3.4) Let $\{X_n,n\ge 1\}$ be a sequence of $\varphi$-mixing random variables. Suppose that p and q are two positive integers. Set $\eta _l=:\sum \nolimits _{j=(l-1)(p+q)+1}^{(l-1)(p+q)+p}X_j$ for $1\le l\le k$. Then

$$\begin{aligned} \left| E\exp \left\{ it\sum _{l=1}^{k}\eta _l\right\} -\prod _{l=1}^k E\exp (it\eta _l)\right| \le C|t|\varphi (q)\sum _{l=1}^k E|\eta _l|. \end{aligned}$$

Proof of Lemma 5.1

It follows from the definition of $S_{1n}^{''}$, Lemma A.4, $Ee_0^2<\infty$ and ($A_1$)–($A_4$) that

$$\begin{aligned} E(S_{1n}^{''})^2\le & {} \sum _{m=1}^{k}\sum _{i=l_m}^{l_m+q-1}E Z_{ni}^2\nonumber \\\le & {} C\sum _{m=1}^{k}\sum _{i=l_m}^{l_m+q-1}\sigma _n^{-2}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}w_{nj}(x)\psi _{j-i}\right) ^2\nonumber \\\le & {} C\sum _{m=1}^{k}\sum _{i=l_m}^{l_m+q-1} \omega _n\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}|\psi _{j-i}|\right) ^2\nonumber \\\le & {} Ckq\omega _n\le Cnqp^{-1}\omega _n=C\gamma _{1n}. \end{aligned}$$

(6.1)

Similar to (6.1), we have

$$\begin{aligned} E(S_{1n}^{'''})^2= & {} E\left( \sum _{i=k(p+q)+1-n}^{2n}Z_{ni}\right) ^2\nonumber \\\le & {} \sum _{i=k(p+q)+1-n}^{2n}E Z_{ni}^2\nonumber \\\le & {} C(3n-k(p+q))\omega _n\le C p\omega _n=C\gamma _{2n}, \end{aligned}$$

(6.2)

and

$$\begin{aligned} ES_{2n}^2= & {} E\left( \sigma _n^{-1}\sum _{i=1}^n w_{ni}(x)\sum _{j: |j|>n}\psi _j e_{i-j}\right) ^2\nonumber \\= & {} E\left| \sigma _n^{-1}\sum _{i_1=1}^n w_{ni_1}(x)\sum _{j_1: |j_1|>n}\psi _{j_1} e_{i_1-j_1}\right| \left| \sigma _n^{-1}\sum _{i_2=1}^n w_{ni_2}(x)\sum _{j_2: |j_2|>n}\psi _{j_2} e_{i_2-j_2}\right| \nonumber \\\le & {} CE\left\{ \sum _{i_1=1}^n\sum _{i_2=1}^n |w_{ni_2}(x)|\cdot \left| \sum _{j_1: |j_1|>n}\psi _{j_1} e_{i_1-j_1}\right| \cdot \left| \sum _{j_2: |j_2|>n}\psi _{j_2} e_{i_2-j_2}\right| \right\} \nonumber \\\le & {} Cn\left( \sum _{j: |j|>n}|\psi _{j}|\right) ^2=C\gamma _{3n}. \end{aligned}$$

(6.3)

This completes the proof of Lemma 5.1. $\square$

Proof of Lemma 5.2

It is easy to see that

$$\begin{aligned} E(S_{1n}^{'})^2= & {} E\left( \sum _{m=1}^{k}y_{nm}\right) ^2\\= & {} \sum _{m=1}^{k}E y_{nm}^2+2\sum _{1\le i<j \le k}\text {Cov}(y_{ni},y_{nj})\\=: & {} s_n^2+2\Gamma _n, \end{aligned}$$

thus,

$$\begin{aligned} |s_n^2-1|\le |E(S_{1n}^{'})^2-1|+2|\Gamma _n|. \end{aligned}$$

(6.4)

Noting that $ES_n^2=1$, we have

$$\begin{aligned} E(S_{1n}^{'})^2= & {} E[S_n-(S_{1n}^{''}+S_{1n}^{'''}+S_{2n})]^2\\= & {} 1+E(S_{1n}^{''}+S_{1n}^{'''}+S_{2n})^2-2ES_n(S_{1n}^{''}+S_{1n}^{'''}+S_{2n}). \end{aligned}$$

Hence, by $C_r$ inequality, Hölder’s inequality, $ES_n^2=1$ and Lemma 5.1, we have

$$\begin{aligned} |E(S_{1n}^{'})^2-1|\le & {} E(S_{1n}^{''}+S_{1n}^{'''}+S_{2n})^2+2E|S_n(S_{1n}^{''}+S_{1n}^{'''}+S_{2n})|\nonumber \\\le & {} E(S_{1n}^{''}+S_{1n}^{'''}+S_{2n})^2+2\left( E|S_nS_{1n}^{''}|+E|S_nS_{1n}^{'''}|+E|S_nS_{2n}|\right) \nonumber \\\le & {} 3\left[ E(S_{1n}^{''})^2+E(S_{1n}^{'''})^2+ES_{2n}^2\right] \nonumber \\&+\,2\left[ \left( E(S_{1n}^{''})^2\right) ^{1/2}+\left( E(S_{1n}^{'''})^2\right) ^{1/2}+\left( ES_{2n}^2\right) ^{1/2}\right] \nonumber \\\le & {} C(\gamma _{1n}^{1/2}+\gamma _{2n}^{1/2}+\gamma _{3n}^{1/2}). \end{aligned}$$

(6.5)

From Lemma A.2, ($A_1$), ($A_4$) and ($A_2$), we have

$$\begin{aligned} |\Gamma _n|\le & {} \sum _{1\le i<j \le k}|\text {Cov}(y_{ni},y_{nj})|\nonumber \\\le & {} \sum _{1\le i<j \le k}\sum _{s=k_i}^{k_i+p-1}\sum _{t=k_j}^{k_j+p-1}|\text {Cov}(Z_{ns},Z_{nt})|\nonumber \\\le & {} \sum _{i=1}^{k-1}\sum _{s=k_i}^{k_i+p-1}\sum _{j=i+1}^{k}\sum _{t=k_j}^{k_j+p-1}\sum _{u=\max \{1,s-n\}}^{\min \{n,s+n\}}\sum _{v=\max \{1,t-n\}}^{\min \{n,t+n\}}\sigma _n^{-2}|w_{nu}(x)w_{nv}(x)|\cdot |\psi _{u-s}\psi _{v-t}|\cdot |\text {Cov}(e_s,e_t)|\nonumber \\\le & {} C\sum _{i=1}^{k-1}\sum _{s=k_i}^{k_i+p-1}\sum _{u=\max \{1,s-n\}}^{\min \{n,s+n\}}|w_{nu}(x)|\cdot |\psi _{u-s}|\sum _{j=i+1}^{k}\sum _{t=k_j}^{k_j+p-1}\varphi ^{1/2}(t-s)\sum _{v=\max \{1,t-n\}}^{\min \{n,t+n\}}|\psi _{v-t}|\nonumber \\\le & {} C\sum _{i=1}^{k-1}\sum _{s=k_i}^{k_i+p-1}\sum _{u=\max \{1,s-n\}}^{\min \{n,s+n\}}|w_{nu}(x)|\cdot |\psi _{u-s}|\cdot \sum _{t:|t-s|\ge q}\varphi ^{1/2}(t-s)\nonumber \\\le & {} C u(q)\sum _{i=1}^{k-1}\sum _{s=k_i}^{k_i+p-1}\sum _{u=1}^{n}|w_{nu}(x)|\cdot |\psi _{u-s}|\nonumber \\\le & {} Cu(q). \end{aligned}$$

(6.6)

Therefore, the desired result follows from (6.5) and (6.6) immediately. $\square$

Proof of Lemma 5.3

By Lemma A.3, we have

$$\begin{aligned} \sup _{u}\left| P\left( T_n/s_n\le u\right) -\Phi (u)\right| \le C\sum _{m=1}^k E|\zeta _{nm}|^{2+\delta }/s_n^{2+\delta }. \end{aligned}$$

(6.7)

According to Lemma A.4, ($A_1$), ($A_2$) and ($A_4$), it follows that

$$\begin{aligned}&\sum _{m=1}^k E|\zeta _{nm}|^{2+\delta }\\&\quad =\sum _{m=1}^k E|y_{nm}|^{2+\delta }=\sum _{m=1}^k E\left| \sum _{i=k_m}^{k_m+p-1}Z_{ni}\right| ^{2+\delta }\\&\quad \le C\sum _{m=1}^k\left\{ \sum _{i=k_m}^{k_m+p-1}E|Z_{ni}|^{2+\delta }+\left( \sum _{i=k_m}^{k_m+p-1}EZ_{ni}^2\right) ^{1+\delta /2}\right\} \\&\quad \le C\sum _{m=1}^k\left\{ \sum _{i=k_m}^{k_m+p-1}\sigma _n^{-(2+\delta )} \left| \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}w_{nj}(x)\psi _{j-i}\right| ^{2+\delta }E|e_i|^{2+\delta }\right. \\&\qquad +\,\left. \left( \sum _{i=k_m}^{k_m+p-1}\sigma _n^{-2}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}w_{nj}(x)\psi _{j-i}\right) ^2 Ee_i^2\right) ^{1+\delta /2}\right\} \\&\quad \le C\sigma _n^{-(2+\delta )} \sum _{m=1}^k\sum _{i=k_m}^{k_m+p-1}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}|w_{nj}(x)||\psi _{j-i}|\right) \cdot \left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}|w_{nj}(x)||\psi _{j-i}|\right) ^{1+\delta }\\&\qquad +\,C\sigma _n^{-(2+\delta )}\sum _{m=1}^k\left( \sum _{i=k_m}^{k_m+p-1}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}w_{nj}(x)\psi _{j-i}\right) ^2\right) \\&\qquad \cdot \,\left( \sum _{i=k_m}^{k_m+p-1}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}w_{nj}(x)\psi _{j-i}\right) ^2\right) ^{\delta /2}\\&\quad \le C\sigma _n^{-(2+\delta )}\cdot \omega _n^{1+\delta }\sum _{j=1}^n|w_{nj}(x)|\sum _{m=1}^k\sum _{i=k_m}^{k_m+p-1}|\psi _{j-i}|\\&\qquad +\,C\sigma _n^{-(2+\delta )}\cdot \omega _n^{1+\delta }\cdot p^{\delta /2}\sum _{j=1}^n|w_{nj}(x)|\sum _{m=1}^k\sum _{i=k_m}^{k_m+p-1}|\psi _{j-i}|\\&\quad \le C\left( \omega _n^{\delta /2}+(p\omega _n)^{\delta /2}\right) \le C(p\omega _n)^{\delta /2}=C\gamma _{2n}^{\delta /2}. \end{aligned}$$

Combined with (6.7) and Lemma 5.2, the proof of Lemma 5.3 is completed. $\square$

Proof of Lemma 5.4

Let $\phi (t)$ and $\psi (t)$ be the characteristic functions of $S_{1n}^{'}$ and $T_n$, respectively. By Esseen inequality (see Petrov 1995, Theorem 5.3), for any $T>0$, we have

$$\begin{aligned}&\sup \limits _{u}|P(S_{1n}^{'}\le u)-P(T_n\le u)|\nonumber \\&\quad \le \int _{-T}^{T}\left| \frac{\phi (t)-\psi (t)}{t}\right| dt+T\sup \limits _{u}\int _{|t|\le C/T}|P(T_{n}\le u+t)-P(T_n\le u)|dt\nonumber \\&\quad =:L_{1n}+L_{2n}. \end{aligned}$$

(6.8)

It can be found by Lemma A.5, ($A_1$), ($A_2$) and ($A_4$) that

$$\begin{aligned} |\phi (t)-\psi (t)|= & {} \left| E\exp \{it S_{1n}^{'}\}-\prod _{m=1}^k E\exp \{it y_{nm}\}\right| \\\le & {} C|t|\varphi (q)\sum _{m=1}^k E|y_{nm}|\\\le & {} C|t|\varphi (q)\sum _{m=1}^k \sum _{i=k_m}^{k_m+p-1}\sigma _n^{-1}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}|w_{nj}(x)\psi _{j-i}|\right) E|e_i|\\\le & {} Ckp\varphi (q)\omega _n^{1/2}|t|=C\gamma _{4n}|t|. \end{aligned}$$

Thus, $L_{1n}\le C\gamma _{4n}T$. It is easily seen by Lemmas 5.3 and 5.2 that

$$\begin{aligned}&\sup \limits _{u}|P(T_{n}\le u+t)-P(T_n\le u)|\\&\quad \le \sup \limits _{u}|P(T_{n}/s_n\le (u+t)/s_n)-\Phi ((u+t)/s_n)|+\sup \limits _{u}|P(T_{n}/s_n\le u/s_n)-\Phi (u/s_n)|\\&\qquad +\,\sup \limits _{u}|\Phi ((u+t)/s_n)-\Phi (u/s_n)|\\&\quad \le C\left( \gamma _{2n}^{\delta /2}+\frac{|t|}{s_n}\right) \le C\left( \gamma _{2n}^{\delta /2}+|t|\right) . \end{aligned}$$

Hence, $L_{2n}\le C(\gamma _{2n}^{\delta /2}+\frac{1}{T}).$ By choosing $T=\gamma _{4n}^{-1/2}$, we can obtain that

$$\begin{aligned} \sup \limits _{u}|P(S_{1n}^{'}\le u)-P(T_n\le u)|\le C\left( \gamma _{2n}^{\delta /2}+\gamma _{4n}^{1/2}\right) . \end{aligned}$$

This completes the proof of the lemma. $\square$

Proof of Lemma 5.5

It follows by Lemma A.4, ($A_1$)–($A_4$) that

$$\begin{aligned} E|S_{1n}^{''}|^{2+\delta }= & {} E\left| \sum _{m=1}^k \sum _{i=l_m}^{l_m+q-1}Z_{ni}\right| ^{2+\delta }\nonumber \\\le & {} C\left[ \sum _{m=1}^k \sum _{i=l_m}^{l_m+q-1}E|Z_{ni}|^{2+\delta }+\left( \sum _{m=1}^k \sum _{i=l_m}^{l_m+q-1}E Z_{ni}^2\right) ^{1+\delta /2}\right] \nonumber \\= & {} C\left[ \sum _{m=1}^k \sum _{i=l_m}^{l_m+q-1} \sigma _n^{-(2+\delta )}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}\omega _{nj}(x)\psi _{j-i}\right) ^{2+\delta }E|e_i|^{2+\delta }\right. \nonumber \\&\left. +\left( \sum _{m=1}^k \sum _{i=l_m}^{l_m+q-1}\sigma _n^{-2}\left( \sum _{j=\max \{1,i-n\}}^{\min \{n,i+n\}}\omega _{nj}(x)\psi _{j-i}\right) ^2 E e_i^2\right) ^{1+\delta /2}\right] \nonumber \\\le & {} C\left[ kq\omega _n^{1+\delta /2}+(k q w_n)^{1+\delta /2}\right] \nonumber \\\le & {} C(k q w_n)^{1+\delta /2}\le C(np^{-1}q w_n)^{1+\delta /2}=C\gamma _{1n}^{1+\delta /2}. \end{aligned}$$

(6.9)

Similarly, we have

$$\begin{aligned} E|S_{1n}^{'''}|^{2+\delta }\le C\gamma _{2n}^{1+\delta /2}. \end{aligned}$$

(6.10)

Thus, according to Markov’s inequality, (6.9), (6.10) and (6.3), we have

$$\begin{aligned} P(|S_{1n}^{''}|>\mu _n)\le \frac{E|S_{1n}^{''}|^{2+\delta }}{\mu _n^{2+\delta }}\le C\mu _n, \\ P(|S_{1n}^{'''}|>\nu _n)\le \frac{E|S_{1n}^{'''}|^{2+\delta }}{\nu _n^{2+\delta }}\le C\nu _n, \end{aligned}$$

and

$$\begin{aligned} P(|S_{2n}|>\lambda _n)\le \frac{ES_{2n}^{2}}{\lambda _n^2}\le C\lambda _n. \end{aligned}$$

The proof is completed. $\square$