On consistency of wavelet estimator in nonparametric regression models

Abstract

In this paper, we mainly investigate the nonparametric regression model with repeated measurements based on extended negatively dependent (END, in short) errors. Based on the Rosenthal type inequality and the Marcinkiewicz–Zygmund type strong law of large numbers, the mean consistency, weak consistency, strong consistency, complete consistency and strong convergence rate of the wavelet estimator are established under some mild conditions, which generalize the corresponding ones for negatively associated errors. Some numerical simulations are presented to verify the validity of the theoretical results.

Appendix

Proof of Corollary 3.1

It is easily checked that

$$\begin{aligned} E\left| V^{(j)}(x)\right| ^q= & {} E\left| \sum _{i=1}^ne^{(j)}(x_i)\int _{A_i}E_k(x,s)ds\right| ^q\nonumber \\\le & {} E\left( \sum _{i=1}^n\frac{\int _{A_i}|E_k(x,s)|ds}{\int _{0}^1|E_k(x,s)|ds}\cdot |e^{(j)}(x_i)|\int _{0}^1|E_k(x,s)|ds\right) ^q\nonumber \\\le & {} \sum _{i=1}^n\frac{\int _{A_i}|E_k(x,s)|ds}{\int _{0}^1|E_k(x,s)|ds}E\left( |e^{(j)}(x_i)|\int _{0}^1|E_k(x,s)|ds\right) ^q\nonumber \\\le & {} \sup _{j\ge 1,0\le x\le 1}E\left| e^{(j)}(x)\right| ^q\cdot \left( \sup _{0\le x\le 1}\int _{0}^1|E_k(x,s)|ds\right) ^q\nonumber \\\le & {} C\sup _{j\ge 1,0\le x\le 1}E\left| e^{(j)}(x)\right| ^q, \end{aligned}$$

(4.1)

where the last inequality above follows by Lemma 3.1 (ii).

Noting that

$$\begin{aligned} V^{(j)}(x)= & {} \sum _{i=1}^ne^{(j)}(x_i)\int _{A_i}E_k(x,s)ds\\= & {} \sum _{i=1}^ne^{(j)}(x_i)\left( \int _{A_i}E_k(x,s)ds\right) ^+-\sum _{i=1}^ne^{(j)}(x_i)\left( \int _{A_i}E_k(x,s)ds\right) ^-\\\doteq & {} V^{(j)}_+(x)-V^{(j)}_-(x), \end{aligned}$$

we have by $C_r$-inequality that

$$\begin{aligned} E\left| \sum _{j=1}^mV^{(j)}(x)\right| ^q\le CE\left| \sum _{j=1}^mV^{(j)}_+(x)\right| ^q+CE\left| \sum _{j=1}^mV^{(j)}_-(x)\right| ^q. \end{aligned}$$

(4.2)

By Lemma 3.3, we can see that $\{V^{(1)}_+(x),V^{(2)}_+(x),\ldots ,V^{(m)}_+(x)\}$ are still zero mean END random variables. Hence, it follows by Lemma 3.4 and (4.1) that

$$\begin{aligned} E\left| \sum _{j=1}^mV^{(j)}_+(x)\right| ^q\le & {} C_q \left[ \sum _{j=1}^mE\left| V^{(j)}_+(x)\right| ^q+\left( \sum _{j=1}^mE\left| V^{(j)}_+(x)\right| ^2\right) ^{q/2}\right] \nonumber \\\le & {} C_q \left[ \sum _{j=1}^mE\left| V^{(j)}(x)\right| ^q+\left( \sum _{j=1}^mE\left| V^{(j)}(x)\right| ^2\right) ^{q/2}\right] \nonumber \\\le & {} C_q\left[ m\sup _{j\ge 1,0\le x\le 1}E\left| e^{(j)}(x)\right| ^q\nonumber \right. \\&\left. +\,m^{q/2}\left( \sup _{j\ge 1,0\le x\le 1}E\left| e^{(j)}(x)\right| ^2\right) ^{q/2}\right] . \end{aligned}$$

(4.3)

Similarly, we have

$$\begin{aligned} E\left| \sum _{j=1}^mV^{(j)}_-(x)\right| ^q\le & {} C_q\left[ m\sup _{j\ge 1,0\le x\le 1}E\left| e^{(j)}(x)\right| ^q\nonumber \right. \\&\left. +\,m^{q/2}\left( \sup _{j\ge 1,0\le x\le 1}E\left| e^{(j)}(x)\right| ^2\right) ^{q/2}\right] . \end{aligned}$$

(4.4)

Therefore, the desired result (3.23) follows by (4.2)–(4.4) immediately. This completes the proof of the corollary. $\square$

Proof of Lemma 3.5

Noting that $\alpha p>1$, we take a suitable q such that $\frac{1}{ \alpha p }< q<1$. For fixed $n\ge 1$, denote for $1\le i\le n$ that

$$\begin{aligned} X_{ni} ^{(1)}= & {} - n^{\alpha q} I(X_{i}<-n^{\alpha q})+X_{i} I(|X_{i} |\le n^{\alpha q}) + n^{\alpha q} I(X_{i}>n^{\alpha q}),\\ X_{ni} ^{(2)}= & {} ( X_{i} -n^{\alpha q} )I(X_{i} >n^{\alpha q}), \\ X_{ni} ^{(3)}= & {} (X_{i} +n^{\alpha q} ) I(X_{i} <-n^{\alpha q}). \end{aligned}$$

Noting that

$$\begin{aligned} \sum _{i=1}^jX_i=\sum _{i=1}^jX_{ni} ^{(1)}+\sum _{i=1}^jX_{ni} ^{(2)}+\sum _{i=1}^jX_{ni} ^{(3)} \end{aligned}$$

for $1\le j\le n$, we have that for all $\varepsilon >0$,

$$\begin{aligned} \sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \max _{1\le j \le n} \left| \sum _{i=1}^{j} X_{i} \right|>\varepsilon n^{\alpha } \right)\le & {} \sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \max _{1\le j \le n} \left| \sum _{i=1}^{j} X_{ni} ^{(1)} \right|> \frac{\varepsilon n^{\alpha }}{3} \right) \nonumber \\&+\sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \max _{1\le j \le n} \left| \sum _{i=1}^{j} X_{ni} ^{(2)} \right|> \frac{\varepsilon n^{\alpha }}{3} \right) \nonumber \\&+\sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \max _{1\le j \le n} \left| \sum _{i=1}^{j} X_{ni} ^{(3)} \right| > \frac{\varepsilon n^{\alpha }}{3} \right) \nonumber \\\doteq & {} I_{1}+ I_{2}+I_3. \end{aligned}$$

(4.5)

Hence, in order to prove (3.2), it suffices to show that $I_1<\infty$, $I_2<\infty$ and $I_3<\infty$.

For $I_1$, we firstly show that

$$\begin{aligned} n^{-\alpha } \max _{1\le j \le n} \left| \sum _{i=1}^{j} E X_{ni} ^{(1)} \right| \rightarrow 0,~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

(4.6)

It follows by $EX_{n} = 0$, Markov’s inequality and Property 1.1 that

$$\begin{aligned} n^{-\alpha } \max _{1\le j \le n} \left| \sum _{i=1}^{j} E X_{ni} ^{(1)} \right|\le & {} n^{-\alpha } \sum _{i=1}^{n} [E |X_{i}| I ( |X_{i}|> n^{\alpha q}) + n^{\alpha q} P( |X_{i}|> n^{\alpha q})] \\\le & {} C n^{-\alpha } \sum _{i=1}^{n} [E |X| I ( |X|> n^{\alpha q}) + n^{\alpha q} P( |X|> n^{\alpha q})] \\\le & {} C n^{-\alpha +1+\alpha q-\alpha pq} E |X|^p I ( |X|> n^{\alpha q}) \\&+\, Cn^{-\alpha +1+\alpha q-\alpha pq} E |X|^{p} \\\le & {} Cn^{-\alpha +1+\alpha q-\alpha pq} E |X|^{p}, \end{aligned}$$

which together with $E |X|^{p} <\infty$ and $\frac{1}{ \alpha p }< q<1$ yields (4.6). Hence, we have by (4.6) that

$$\begin{aligned} I_{1} \le C\sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \max _{1\le j \le n} \left| \sum _{i=1}^{j} (X_{ni} ^{(1)} - EX_{ni} ^{(1)}) \right| >\frac{\varepsilon n^{\alpha }}{6} \right) . \end{aligned}$$

(4.7)

For fixed $n\ge 1$, we can see that $\{X_{ni} ^{(1)}-EX_{ni} ^{(1)}, 1\le i\le n\}$ are still END random variables by Lemma 3.3. It follows by (4.7), Markov’s inequality and Lemma 3.4 that for any $\delta \ge 2$,

$$\begin{aligned} I_{1}\le & {} C \sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } E \left( \max _{1\le j \le n} \left| \sum _{i=1}^{j} \left( X_{ni} ^{(1)} - EX_{ni} ^{(1)}\right) \right| \right) ^{\delta } \nonumber \\\le & {} C \sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } \log ^{\delta } n\left[ \sum _{i=1}^{n} E\left| X_{ni} ^{(1)}\right| ^{\delta } + \left( \sum _{i=1}^{n} E\left| X_{ni} ^{(1)}\right| ^{2} \right) ^{\delta /2} \right] \nonumber \\\doteq & {} CI_{11} +CI_{12}. \end{aligned}$$

(4.8)

Taking $\delta >\max \left\{ \frac{\alpha p-1}{\alpha -1/2}, 2, p\right\}$, we have that

$$\begin{aligned} \alpha p-1-\alpha \delta + \alpha q\delta -\alpha pq=\alpha (p-\delta )(1-q)-1<-1 \end{aligned}$$

and

$$\begin{aligned} \alpha p-2-\alpha \delta +\frac{\delta }{2}<-1. \end{aligned}$$

It follows by $C_r$-inequality, Markov’s inequality and Property 1.1 that

$$\begin{aligned} I_{11}\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta }\log ^{\delta } n \sum _{i=1}^{n} [E |X_{i}|^{\delta } I ( |X_{i}| \le n^{\alpha q}) + n^{\alpha q \delta } P( |X_{i}|> n^{\alpha q})] \nonumber \\\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-1-\alpha \delta } \log ^{\delta } n [E |X|^{\delta } I ( |X| \le n^{\alpha q}) + n^{\alpha q \delta } P( |X| > n^{\alpha q})] \nonumber \\\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-1-\alpha \delta + \alpha q\delta -\alpha pq} E|X|^p \log ^{\delta } n \nonumber \\< & {} \infty \end{aligned}$$

(4.9)

and

$$\begin{aligned} I_{12}\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta }\log ^{\delta } n \left\{ \sum _{i=1}^{n} \left[ E X_{i}^{2} I ( |X_{i}| \le n^{\alpha q}) + n^{2\alpha q} P( |X_{i}|> n^{\alpha q})\right] \right\} ^{\delta /2} \nonumber \\\le & {} C \sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta +\frac{\delta }{2} } \log ^{\delta } n [E X^{2} I ( |X| \le n^{\alpha q}) + n^{2\alpha q} P( |X| > n^{\alpha q})] ^{\delta /2} \nonumber \\\le & {} \left\{ \begin{array}{lll} C\sum \limits _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta +\frac{\delta }{2} } (E X^{2}) ^{\delta /2} \log ^{\delta } n ,&{}\quad \text {if}~~p\ge 2\\ C\sum \limits _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta +\frac{\delta }{2}+\alpha (2-p) \frac{\delta }{2} } (E |X|^{p}) ^{\delta /2} \log ^{\delta } n,&{}\quad \text {if}~~1\le p<2\\ \end{array}\right. \nonumber \\= & {} \left\{ \begin{array}{lll} C\sum \limits _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta +\frac{\delta }{2} } (E X^{2}) ^{\delta /2} \log ^{\delta } n,&{}\quad \text {if}~~p\ge 2\\ C\sum \limits _{n=1}^{\infty } n^{(\alpha p-1)(1-\delta /2)-1 } (E |X|^{p}) ^{\delta /2} \log ^{\delta } n,&{}\quad \text {if}~~1\le p<2\\ \end{array}\right. \nonumber \\< & {} \infty . \end{aligned}$$

(4.10)

Hence, $I_1<\infty$ follows by (4.8)–(4.10) immediately.

In the following, we will show that $I_{2}<\infty$. For fixed $n\ge 1$, denote for $1\le i\le n$ that

$$\begin{aligned} X_{ni} ^{(4)} = (X_{i}- n^{\alpha q} ) I (n^{\alpha q} < X_{i} \le n^{\alpha }+ n^{\alpha q} ) + n^{\alpha } I ( X_{i} > n^{\alpha }+ n^{\alpha q} ). \end{aligned}$$

It is easily checked that

$$\begin{aligned} \left( \max _{1\le j\le n}\left| \sum _{i=1}^{j} X_{ni} ^{(2)}\right|> \frac{\varepsilon n^{\alpha }}{3} \right) \subset \left( \max _{1\le i\le n}X_{i}> n^{\alpha }\right) \bigcup \left( \max _{1\le j\le n}\left| \sum _{i=1}^{j} X_{ni} ^{(4)}\right| > \frac{\varepsilon n^{\alpha }}{3} \right) , \end{aligned}$$

which implies that

$$\begin{aligned} I_{2}\le & {} \sum _{n=1}^{\infty } n^{\alpha p-2} \sum _{i=1}^{n} P(|X_{i}|> n^{\alpha }) + \sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \max _{1\le j\le n}\left| \sum _{i=1}^{j} X_{ni} ^{(4)}\right| > \frac{\varepsilon n^{\alpha }}{3} \right) \nonumber \\\doteq & {} I_{21}+ I_{22}. \end{aligned}$$

(4.11)

It follows by $E|X|^{p} < \infty$ that

$$\begin{aligned} I_{21} \le C\sum _{n=1}^{\infty } n^{\alpha p-1} P(|X|> n^{\alpha }) \le C E|X|^{p} < \infty . \end{aligned}$$

(4.12)

Noting that $\frac{1}{\alpha p}<q<1$, we have by the definition of $X_{ni} ^{(4)}$ and Property 1.1 that

$$\begin{aligned} n^{-\alpha } \max _{1\le j \le n} \left| \sum _{i=1}^{j} E X_{ni} ^{(4)} \right|\le & {} C n^{1-\alpha } E|X|I(|X|> n^{\alpha q}) \nonumber \\\le & {} Cn ^{1-\alpha +\alpha q- \alpha pq } E|X|^{p} \rightarrow 0,~~\text {as}~~n\rightarrow \infty . \end{aligned}$$

(4.13)

Since $X_{ni} ^{(4)}>0$, we have by (4.11)–(4.13) that

$$\begin{aligned} I_{2}\le C\sum _{n=1}^{\infty } n^{\alpha p-2} P \left( \left| \sum _{i=1}^{n} \left( X_{ni} ^{(4)} - EX_{ni} ^{(4)} \right) \right| >\frac{\varepsilon n^{\alpha }}{6} \right) . \end{aligned}$$

(4.14)

For fixed $n\ge 1$, we can see that $\{X_{ni} ^{(4)}-EX_{ni} ^{(4)}, 1\le i\le n\}$ are still END random variables by Lemma 3.3. It follows by Markov’s inequality, $C_r$ inequality and Lemma 3.4 that

$$\begin{aligned} I_{2}\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } E \left| \sum _{i=1}^{n} \left( X_{ni} ^{(4)} - EX_{ni} ^{(4)} \right) \right| ^{\delta } \nonumber \\\le & {} C \sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } \left[ \sum _{i=1}^{n} E|X_{ni} ^{(4)}|^{\delta } + \left( \sum _{i=1}^{n} E(X_{ni} ^{(4)})^{2} \right) ^{\delta /2} \right] \nonumber \\\doteq & {} J_{1}+ J_{2}. \end{aligned}$$

(4.15)

By $C_r$-inequality and Property 1.1 again, we can get that

$$\begin{aligned} J_{1}\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } \sum _{i=1}^{n} \left[ E |X_{i}- n^{\alpha q}|^{\delta } I (n^{\alpha q}< X_{i} \le n^{\alpha }\right. \nonumber \\&\left. +\, n^{\alpha q} ) + n^{\alpha \delta } P ( X_{i}> n^{\alpha }+ n^{\alpha q} ) \right] \nonumber \\\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } \sum _{i=1}^{n} \left[ E |X_{i}|^{\delta } I(|X_{i}| \le 2 n^{\alpha }) + n^{\alpha \delta } P ( X_{i}> n^{\alpha } )\right] \nonumber \\\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-2-\alpha \delta } \sum _{i=1}^{n}\left[ E |X|^{\delta } I(|X| \le 2 n^{\alpha }) + n^{\alpha \delta } P ( |X |> n^{\alpha } ) \right] \nonumber \\\le & {} C\sum _{n=1}^{\infty } n^{\alpha p-1-\alpha \delta } E |X|^{\delta } I(|X| \le 2 n^{\alpha }) +CE|X|^p \nonumber \\= & {} C\sum _{n=1}^{\infty } n^{\alpha p-1-\alpha \delta } \sum _{i=1}^nE |X|^{\delta } I(2(i-1)^\alpha<|X| \le 2 i^{\alpha }) +CE|X|^p \nonumber \\\le & {} C \sum _{i=1}^\infty i^{\alpha \delta } P(2(i-1)^\alpha<|X| \le 2 i^{\alpha })\sum _{n=i}^{\infty } n^{\alpha p-1-\alpha \delta } +CE|X|^p \nonumber \\\le & {} C \sum _{i=1}^\infty i^{\alpha p} P(2(i-1)^\alpha<|X| \le 2 i^{\alpha }) +CE|X|^p \nonumber \\\le & {} CE|X|^p ~< ~\infty . \end{aligned}$$

(4.16)

Similar to the proofs of (4.10) and (4.16), we can obtain that $J_2<\infty$, which together with (4.15) and (4.16) yields that $I_2<\infty$.

Similar to the proof of $I_2<\infty$, one can get that $I_3<\infty$. Hence, (3.2) follows by (4.5) and $I_1<\infty$, $I_2<\infty$ and $I_3<\infty$ immediately. By the standard method, one can get (3.3) by (3.2) immediately. This completes the proof of the theorem. $\square$