Asymptotics of a wavelet estimator in the nonparametric regression model with repeated measurements under a NA error process

Abstract

Consider the nonparametric regression model with repeated measurements: $Y^{(j)}(x_{ni})=g(x_{ni})+e^{(j)}(x_{ni})$, where $Y^{(j)}(x_{ni})$ is the $j$th response at the point $x_{ni}$, $x_{ni}$’s are known and nonrandom, and $g(\cdot )$ is unknown function defined on a closed interval $[0,1]$. For exhibiting the correlation among the units and avoiding any assumptions among the observations within the same unit, we consider the model with negative associated (NA) error structures, that is, $\{e^{(j)}(x), j\ge 1\}$ is a mean zero NA error process. The wavelet procedures are developed to estimate the regression function. Some asymptotics of wavelet estimator are established under suitable conditions.

Appendix

In this section, we given some preliminary Lemmas, which have been used in Sect. 3.

Lemma 4.1

([1, 24]) Suppose that $($A3$)($iii$)$ holds. We have

• (a) $\sup _{0\le x,s \le 1}|E_k(x,s)|=O(2^k)$

• (b) $\sup _{0\le x\le 1}\int _0^1|E_k(x,s)|ds\le C$

• (c) $\int _0^1 E_k(x,s)ds\rightarrow 1$ uniformly in $x\in [0,1]$, as $k\rightarrow \infty$

Lemma 4.2

([1]) Suppose that $($A3$)($iii$)($iv$)$ hold, and $h(x)$ satisfies $($A3$)($i$)($ii$)$. Then

$$\begin{aligned} \sup _{x\in \mathcal {I}}\left| h(x)-\sum _{i=1}^n h(x_i)\int \limits _{A_i}E_k(x,s)ds\right| =O(n^{-\gamma })+O(\tau _k), \end{aligned}$$

where

$$\begin{aligned} \tau _k=\left\{ \begin{array}{l@{\quad }c@{\quad }l} (1/2^k)^{\upsilon -1/2} &{} if &{} 1/2<\upsilon <3/2, \\ \sqrt{k}/2^k &{} if &{} \upsilon =3/2, \\ 1/2^k &{} if &{} \upsilon >3/2.\\ \end{array}\right. \end{aligned}$$

Lemma 4.3

([20]) Let $\{X_i,i\ge 1\}$ be a sequence of NA random variables with $E|X_i|^p<\infty$ for some $p\ge 2$. Then, there exists a constant $C_p>0$ such that

$$\begin{aligned} E\max _{1\le k\le n}\left| \sum _{i=1}^k X_i\right| ^p\le C_p\left\{ \left( \sum _{i=1}^nEX_i^2\right) ^{p/2}+\sum _{i=1}^nE|X_i|^p\right\} . \end{aligned}$$

Lemma 4.4

([13, 21]) Let $\{X_i,i\ge 1\}$ be a sequence of identically distributed NA random variables, if $E|X_1|<\infty$, then

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n(X_i-EX_1)\rightarrow 0, ~a.s.. \end{aligned}$$

Lemma 4.5

([14]) Let $\{X_i,i\ge 1\}$ be a sequence of strictly stationary NA random variables with $EX_1=0$, $0<EX_1^2<\infty$ and

$$\begin{aligned} \sigma ^2=EX_1^2+2\sum _{j=2}^\infty EX_1X_j>0, \end{aligned}$$

then

$$\begin{aligned} \frac{1}{\sigma \sqrt{n}}\sum _{i=1}^nX_i\mathop {\longrightarrow }\limits ^{d}N(0,1). \end{aligned}$$

Lemma 4.6

Let $V^{(j)}(x)=\sum _{i=1}^ne^{(j)}(x_{ni})\int _{A_i}E_k(x,s)ds$. If $\max _{1\le j\le m,1\le i\le n}E|e^{(j)}(x_{ni})|^{1+\delta }<\infty$ for some $\delta >0$, then

$$\begin{aligned} \sup _{x\in \mathcal {I}}E|V^{(j)}(x)|^{1+\delta }<C. \end{aligned}$$

Proof

Note that $f(t)=t^{1+\delta } (t>0)$ is a convex function on real set $\mathbb {R}$. We have

$$\begin{aligned} \sup _{x\in \mathcal {I}}E\left| V^{(j)}(x)\right| ^{1+\delta }&= \sup _{x\in \mathcal {I}}E\left[ \left( \sum _{i=1}^ne^{(j)}(x_{ni})\int \limits _{A_i}E_k(x,s)ds\right) ^{1+\delta }\right] \\&\le \sup _{x\in \mathcal {I}}E\left[ \left( \sum _{i=1}^n|e^{(j)}(x_{ni})|\int \limits _{A_i}|E_k(x,s)|ds\right) ^{1+\delta }\right] \\&= \sup _{x\in \mathcal {I}}E\left[ \left( \sum _{i=1}^n\frac{\int \limits _{A_i}|E_k(x,s)|ds}{\int \limits _{0}^1|E_k(x,s)|ds}\cdot |e^{(j)}(x_{ni})|\int \limits _{0}^1|E_k(x,s)|ds\right) ^{1+\delta }\right] \\&\le \sup _{x\in \mathcal {I}}\sum _{i=1}^n\frac{\int _{A_i}|E_k(x,s)|ds}{\int \limits _{0}^1|E_k(x,s)|ds}\cdot E\left[ \left( |e^{(j)}(x_{ni})|\int \limits _{0}^1|E_k(x,s)|ds\right) ^{1+\delta }\right] \\&\le \sup _{x\in \mathcal {I}}\sum _{i=1}^n\frac{\int \limits _{A_i}|E_k(x,s)|ds}{\int \limits _{0}^1|E_k(x,s)|ds}\cdot E\left( |e^{(j)}(x_{ni})|^{1+\delta }\right) \left( \int \limits _{0}^1|E_k(x,s)|ds\right) ^{1+\delta }\\&\le \max _{1\le i\le n}E|e^{(j)}(x_{ni})|^{1+\delta }\sup _{x\in \mathcal {I}}\left( \int \limits _{0}^1|E_k(x,s)|ds\right) ^{1+\delta }\le C. \end{aligned}$$

$\square$

Lemma 4.7

Assume that $\max _{1\le i\le n}E|e^{(j)}(x_{ni})|^p<\infty$ for each $j (1\le j\le m)$ and some $p\ge 2$. Then there exists a constant $C_p$ such that

$$\begin{aligned} E\max _{1\le l\le m}\left| \sum _{j=1}^lV^{(j)}(x)\right| ^p\le C_p\left\{ \left( \sum _{j=1}^m\max _{1\le i\le n}E|e^{(j)}(x_{ni})|^2\right) ^{p/2}+\sum _{j=1}^m\max _{1\le i\le n}E|e^{(j)}(x_{ni})|^p\right\} . \end{aligned}$$

Proof

Let $a_{ni}(x)=\int _{A_i}E_k(x,s)ds$, then $V^{(j)}(x)=\sum _{i=1}^n e^{(j)}(x_{ni})a_{ni}(x)$. Obviously, $V^{(j)}(x)=\sum _{i=1}^n e^{(j)}(x_{ni})a_{ni}^+(x)-\sum _{i=1}^n e^{(j)}(x_{ni})a_{ni}^-(x)$. Without loss of generality we can assume that $a_{ni}(x)\ge 0$, $i=1,\cdots ,n$, for each $x\in \mathcal {I}$. So $\{V^{(j)}(x), 1\le j\le m\}$ still are zero mean NA random variables. Hence, by using Lemma A.3, and with arguments similar to proof of Lemma A.6, we have

$$\begin{aligned} E\max _{1\le l\le m}\left| \sum _{j=1}^lV^{(j)}(x)\right| ^p&\le C_p\left\{ \left( \sum _{j=1}^mE|V^{(j)}(x)|^2\right) ^{p/2}+\sum _{j=1}^mE|V^{(j)}(x)|^p\right\} \\&\le C_p\left\{ \left( \sum _{j=1}^m\max _{1\le i\le n}E|e^{(j)}(x_{ni})|^2\right) ^{p/2}+\sum _{j=1}^m\max _{1\le i\le n}E|e^{(j)}(x_{ni})|^p\right\} . \end{aligned}$$

Thus, we complete the proof. $\square$

Lemma 4.8

Let $\{X_i,i\ge 1\}$ be a sequence of identically distributed NA random variables, if $E|X_1|^{1+\delta }<\infty$ for some $\delta \ge 0$, then

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n|X_i|^{1+\delta }\rightarrow E|X_1|^{1+\delta },\,a.s.. \end{aligned}$$

Proof

Let $|X_i|^{1+\delta }=(X_i^+)^{1+\delta }+(X_i^-)^{1+\delta }$. Note that $\{(X_i^+)^{1+\delta }, i\ge 1\}$ and $\{(X_i^-)^{1+\delta }, i\ge 1\}$ still are the sequences of identically distributed NA random variables with $E|X_1^+|^{1+\delta }<\infty$ and $E|X_1^-|^{1+\delta }<\infty$, respectively. By Lemma A.4, we have

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n(|X_i^+|^{1+\delta }-E|X_1^+|^{1+\delta })\rightarrow 0, \,a.s., \quad \frac{1}{n}\sum _{i=1}^n(|X_i^-|^{1+\delta }-E|X_1^-|^{1+\delta })\rightarrow 0, \,a.s.. \end{aligned}$$

Hence, we obtain result of Lemma A.8. $\square$