A Berry-Ess$\acute{e}$n bound of wavelet estimation for a nonparametric regression model under linear process errors based on LNQD sequence

1.   Introduction

Consider the classical nonparametric regression model

where $\{\varepsilon_{ni}, 1\leq i\leq n\}$ are random errors and $\{ x_{ni}, 1\leq i\leq n\}$ are known fixed design points, $g(\cdot)$ is an unknown bounded real valued function on $[0, 1].$ It is well known that the model (1.1) has been widely studied by many authors in the literature. The uniformly asymptotic normality for a general weighted regression estimator of $g(\cdot)$, which was proposed by Georgiev ^[1], had got extensive investigated. One can refer to Georgiev^[2] under independent random errors, Roussas et al.^[3] under strong mixing random errors, Yang^[4] under negatively associated errors, and so on. Under the assumption that the errors case is a weakly stationary linear process based on a martingale difference sequence, Tran et al.^[5] studied the asymptotic normality. Liang and Li^[6] obtained the Berry-Ess$\acute{e}$en bound based on linear process errors under negatively associated random variables.

In recent years, wavelets techniques, owing to their ability to adapt to local features of curves, have been widely used in statistics, engineering and technological fields. Many authors have considered employing wavelet methods to estimate nonparametric and semiparametric models. For instance, one can refer to the papers^{[7,8,9,10,11,12,13]}.

The problem we face here is to derive a Berry-Ess$\acute{e}$en bound of wavelet estimator of $g(\cdot)$ proposed by Antoniadis et al.^[7]

where the wavelet kernel $E_m(t, s)$ can be defined by $E_m(t, s) = 2^m\sum_{k\in Z}\phi(2^mt-k)\phi(2^ms-k), \; \phi(\cdot)$ is a scaling function, $m = m(n) > 0$ is an integer depending only on $n$, $A_i = [s_{i-1}, s_i), i = 1, 2, \cdots, n$ are intervals that partition $[0, 1]$.

We recall the concepts of negative associated (NA, in short), negative quadrant dependent (NQD, in short) and linearly negative quadrant dependent (LNQD, in short) sequences.

Definition 1.1. ^[14] A finite collection of random variables $\{X_i\}_{1\leq i\leq n}$ are said to be NA, if for every disjoint subsets $A, B\subset \{1, 2, \cdots, n\}$

where $f$ and $g$ are real coordinate-wise nondecreasing functions such that this covariance exists. An infinite sequence of random variables $\{X_n\}_{n\geq 1}$ are said to be NA, if for every $n\geq 2$, $X_1, X_2, \cdots, X_n$ are NA.

Definition 1.2. ^[15] Two random variables $X, Y$ are said to be NQD, if for any $x, y\in R,$

A sequence of random variables $\{X_n\}_{n\geq 1}$ are said to be pairwise negative quadrant dependent (PNQD, in short), if every pair of random variables in the sequence are NQD.

Definition 1.3. ^[16] A sequence $\{X_i\}_{1\leq i\leq n}$ of random variables are said to be LNQD, if for any disjoint subsets $A, B\subset Z^+$ and positive $r_j's, \; \sum_{i\in A}r_iX_i$ and $\sum_{j\in B}r_jX_j$ are NQD.

Remark 1.1. It easily follows that if $\{X_n\}_{n\geq 1}$ is a sequence LNQD random variables, then $\{aX_n+b\}_{n\geq 1}$ is still a sequence of LNQD, where $a$ and $b$ are real numbers. Furthermore, NA implies LNQD from the definitions, LNQD random variables are NQD random variables, but the converse is not true.

The concept of LNQD sequence was introduced by Newman^[16], some applications can be found in many monographs. For example, Newman investigated the central limit theorem for a strictly stationary LNQD process. Wang et al.^[17] established the exponential inequalities and complete convergence for a LNQD sequence. Li et al.^[18] obtained some inequalities and gave some applications for a fixed-design regression model. Ding et al.^[19] derived the Berry-Ess$\acute{e}$en bound of weighted kernel estimator for model (1.1) based on linear process errors under a LNQD sequence.

In this paper, we shall consider the above wavelet estimator of nonparametric regression problem with linear process errors generated by a LNQD sequence. Our main purpose is to derive the Berry-Ess$\acute{e}$en bound of the wavelet estimator (1.2).

The layout of the rest is as follows. In Section 2, we present some basic assumptions and main results. In Section 3, some preliminary lemmas are stated and proof of Theorem 2.1 is provided. In Section 4, proofs of some preliminary lemmas are given. The Appendix contains some known results.

Throughout the paper, $C, C_1, C_2, \cdots$ denote some positive constants not depending on $n,$ which may be different in various places. $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$. All limits are taken as the sample size $n$ tends to $\infty$, unless specified otherwise.

2.   Some basic assumptions and main results

To obtain our results, the following basic assumptions are sufficient:

Assumption (A1) For each $n$, $\{\varepsilon_{ni}, 1\leq i\leq n\}$ have the same joint distribution as $\{\xi_1, \cdots, \xi_n\}$, where $\xi_t = \sum_{j = -\infty}^{\infty}|a_j|e_{t-j}$ and $\{a_j\}$ is a sequence of real numbers with $\sum_{j = -\infty}^{\infty}|a_j| < \infty.$ Here $\{e_j\}$ are identically distribution, LNQD random variables with $Ee_0 = 0, E|e_0|^{2+\delta} < \infty, 0 < \delta\leq 1.$

Assumption (A2) The spectral density function $f(\omega)$ of $\{\xi_i\}$ is bounded away from zero and infinity, i.e., for $\omega\in (-\pi, \pi], 0 < C_1\leq f(\omega)\leq C_2 < \infty$.

Assumption (A3) (ⅰ)$\phi(\cdot)$ is $r-$ regular ($r$ is a positive integer), and satisfies the Lipschitz condition with order 1 with a compact support. Furthermore, $|\hat{\phi}(\xi)-1| = O(\xi)$ as $\xi\rightarrow\infty,$ where $\hat{\phi}$ denotes the Fourier transform of $\phi.$ (ii) $\max_{1\leq i\leq n}|s_i-s_{i-1}| = O(n^{-1}).$

Assumption (A4) (ⅰ) $g(\cdot)\in H^v, v > 1/2,$ where $H^v$ presents Sobolev space of order $v$, i.e., if $h\in H^v$ then $\int|\hat{h}(\omega)|^2(1+\omega^2)^vd\omega < \infty$ with $\hat{h}$ denoting the Fourier transform of $h$. (ii) $g(\cdot)$ satisfies the Lipschitz condition of order 1.

Assumption (A5) There exist positive integers $p: = p_n, q: = q_n$ and $k: = k_n = \lfloor \frac{3n}{p+q}\rfloor,$ such that, for $p+q\leq 3n, qp^{-1}\rightarrow 0$, and let $\gamma_{in}\rightarrow 0, i = 1, 2, 3$, where $\gamma_{1n} = qp^{-1}2^m, \gamma_{2n} = pn^{-1}2^m, \gamma_{3n} = n(\sum_{|j| > n}|a_j|)^2$.

Remark 2.1. Assumption (A1) is the general condition of the LNQD sequence, Assumptions (A2–A4) are mild regularity conditions for wavelet estimate in the recent literature, such as Sun and Chai^[10], Li et al.^[11,12], Liang and Qi ^[8]. In Assumption (A5) $\gamma_{in}\rightarrow 0, i = 1, 2, 3$ are easily satisfied if $p, q, m$ are chosen reasonable, see e.g., Li et al. ^[11,12] and Liang et al. ^[6,8].

In order to formulate our main results, let $\sigma^2_n: = \sigma^2_n(t) = \mathrm{Var}(\hat{g}_n(t)) > 0, \; S_n: = S_n(t) = \sigma_n^{-1}\{\hat{g}_n(t)-E\hat{g}_n(t)\}, \; u(q) = \sup_{j\geq 1}\sum_{j:|j-i|\geq q}|\mathrm{Cov}(e_i, e_j)|$.

Theorem 2.1. Assume that Assumptions (A1)–(A5) are satisfied, then for each $t\in [0, 1]$, we have

where $\Phi(x)$ is the distribution function of $N(0, 1).$

Remark 2.2. Theorem 2.1 extends the results of Li et al.^[11] from associated samples to linear process errors generated by a LNQD sequence.

Corollary 2.1. Assume that conditions of Theorem 2.1 hold and $u(1) < \infty$, for each $t\in [0, 1],$ then

Corollary 2.2. Under the conditions of Theorem 2.1, let $\delta = \frac{2}{3}, \; n^{-1}{2^m} = O(n^{-\theta}), u(n) = O(n^{-\frac{\theta-\rho}{2\rho-1}})$ and $\sup\limits_{n\geq 1}\left(n^{\frac{\theta-\rho+1}{2}}\right)\sum\limits_{|j| > n}|a_j| < \infty$, where $\frac{1}{2} < \rho\leq\theta < 1$, for each $t\in [0, 1],$ then

Remark 2.3. From Corollary 2.2, taking $\theta\approx 1$ and $\rho\approx \frac{1}{2}$, the rate of convergence is near $O(n^{-1/6}).$

3.   Some preliminary lemmas

In order to prove our main results we introduce the following preliminary lemmas. At first, some notations are introduced for the sake of convenience and brevity.

According to the Eq (1.1) and (1.2), we have

Note that

Set $S_{1n} = S'_{1n}+S''_{1n}+S'''_{1n},$ where $S'_{1n} = \sum_{m = 1}^ky_{nm}, S''_{1n} = \sum_{m = 1}^ky'_{nm}, S'''_{1n} = y'_{nk+1},$ $y_{nm} = \sum_{i = k_m}^{k_m+p-1}Z_{ni}, y'_{nm} = \sum_{i = l_m}^{l_m+q-1}Z_{ni}, y'_{nk+1} = \sum_{i = k(p+q)-n+1}^{2n}Z_{ni}, \; k_m = (m-1)(p+q)+1-n, l_m = (m-1)(p+q)+p+1-n, m = 1, 2, \cdots, k.$ Then we have $S_n = S'_{1n}+S''_{1n}+S'''_{1n}+S_{2n}.$

Lemma 3.1. Assume that (A1)–(A5) are satisfied, then

$(i) E(S''_{1n})^2\leq C\gamma_{1n}, \; E(S'''_{1n})^2\leq C\gamma_{2n}, \; E(S_{2n})^2\leq C\gamma_{3n};$

$(ii) P(|S''_{1n}|\geq\gamma^{1/3}_{1n})\leq C\gamma^{1/3}_{1n}, \; P(|S'''_{1n}|\geq\gamma^{1/3}_{2n})\leq C\gamma^{1/3}_{2n}, \; P(|S_{2n}|\geq\gamma^{1/3}_{3n})\leq C\gamma^{1/3}_{3n}.$

Lemma 3.2. Suppose that (A1)–(A5) hold, let $s_n^2 = \sum_{m = 1}^k \mathrm{Var}(y_{nm})$, then

Let $\{\eta_{nm}:m = 1, 2, \cdots, k\}$ be independent random variables and $\eta_{nm}$ have the same distribution as $y_{nm}, m = 1, 2, \cdots, k.$ Set $H_n = \sum_{m = 1}^k\eta_{nm}.$

Lemma 3.3. Under Assumptions (A1)–(A5), we have

Lemma 3.4. Under the Assumptions of Theorem 2.1, we have

Lemma 3.5. Assume that (A1)–(A5) are true, then

Proof of Theorem 2.1. Similar to the proof of Theorem 2.1 in [6], it is easily seen that

From Lemma 3.2 and Lemma 5.2 in Petrov^[20], it follows that

Consequently, by means of Lemmas 3.2–3.4, we can get

According to Lemma A.1 in the Appendix, Lemma 3.1(ⅱ) and the Eq (3.1), we obtain

This completes the proof of Theorem 2.1.

Proof of Corollary 2.1. According to $u(1) < \infty$ it easily follows that $u(q)\rightarrow 0$, hence Corollary 2.1 holds by Theorem 2.1.

Proof of Corollary 2.2. Taking $p = \lfloor n^{\rho}\rfloor, q = \lfloor n^{2\rho-1}\rfloor$ in Theorem 2.1, for $\delta = 2/3, 1/2 < \rho\leq \theta < 1$, we obtain

Therefore, the conclusion follows from Theorem 2.1.

4.   Proofs of some preliminary lemmas

Proof of Lemma 3.1. According to Lemma 3.5, Lemmas A.2, A.4 in the Appendix and Assumptions (A1), (A5), we have

As to $S_{2n},$ by Lemma A.4 in the Appendix

Therefore the proof of Lemma 3.1(ⅰ) is completed. In addition, by Markov inequality and Lemma 3.1(ⅰ) it easily follows that Lemma 3.1(ⅱ) is true.

Proof of Lemma 3.2. Set $\Gamma_n = \sum\limits_{1\leq i < j\leq n}\mathrm{Cov}(y_{ni}, y_{nj})$, then $s_n^2 = E(S'_{1n})^2-2\Gamma_n.$ Note that $ES_n^2 = 1$ and applying Lemma 3.1(ⅰ), we can get

On the other hand, by Lemma 3.1, Lemma 3.5, Assumptions (A1), (A5), and Lemma A.4 in the Appendix, it follows that

Thus, (4.1) and (4.2) follow that $|s^2_n-1|\leq C(\gamma^{1/2}_{1n}+\gamma^{1/2}_{2n}+\gamma^{1/2}_{3n}+u(q))$.

Proof of Lemma 3.3. Applying the Berry-Ess$\acute{e}$en inequality (cf.Petrov^[20], Theorem 5.7), we get

By Lemma 3.5, from Assumption(A1), (A5) and A.2 in the Appendix it follows that

Since $s_n\rightarrow 1$ by Lemma 3.2. From (4.3) and (4.4) we can get Lemma 3.3.

Proof of Lemma 3.4. Let $\psi(t)$ and $\varphi(t)$ be the characteristic functions of $S'_{1n}$ and $H_n$, respectively. Thus applying the Ess$\acute{e}$en inequality(Petrov^[20], Theorem 5.3), for any $T > 0,$

Similar to (4.2), it follows from Lemma 3.5, A.3 and A.4 in the Appendix that

which implies that

Therefore, by Lemma 3.3, we have

Hence, from (4.7) it follows that

Combining (4.5), (4.6) with (4.8), and choosing $T = u^{-1/3}(q)$, we can easily see that

Proof of Lemma 3.5. By Assumption (A1) and Lemma A.5 in the Appendix, it follows that

In addition, according to Assumptions (A2)–(A4) and referring to Liang and Qi^[8], it is easy to follow Lemma 3.5.

5.   Appendix

Lemma A.1 ^[4] Suppose that $\{\varsigma_n, n\geq 1\}, \{\eta_n, n\geq 1\}$ and $\{\xi_n, n\geq 1\}$ are three random variable sequences, $\{\gamma_n, n\geq 1\}$ is a positive constant sequence, and $\gamma_n\rightarrow 0.$ If $\sup\limits_{x}|F_{\varsigma_n}(x)-\Phi(x)|\leq C\gamma_n,$ then for any $\varepsilon_1 > 0, \varepsilon_2 > 0$

Lemma A.2 ^[18] Let $\{X_j\}_{j\geq 1}$ be a LNQD random variable sequence with zero mean and finite second moment, $\sup_{j\geq 1}EX^r_j < \infty$. Assume that $\{a_j\}_{j\geq 1}$ be a real constant sequence, $a: = \sup\limits_{j}|a_j| < \infty.$ Then for any $r > 1,$ there exists a constant C not depending on $n$ such that

Lemma A.3 ^[18] If $X_1, \cdots, X_m$ are LNQD random variables with finite second moments, let $\varphi_j(t_j)$ and $\varphi(t_1, \cdots, t_m)$ be c.f.'s of $X_j$ and $(X_1, \cdots, X_m),$ respectively, then for all nonnegative(or non positive) real numbers $t_1, \cdots, t_m,$

Lemma A.4 ^[11] Assume that Assumptions (A3) and (A4) hold, then

$(i)\sup_{m}\int_0^1|E_m(t, s)|ds\leq C; \; (ii) \sum_{i = 1}^n|\int_{A_i}E_m(t, s)ds|\leq C;$

$(iii)|\int_{A_i}E_m(t, s)ds| = O(\frac{2^m}{n}), i = 1, 2, \cdots, n; \; (iv) \sum_{i = 1}^n(\int_{A_i}E_m(t, s)ds)^2 = O(\frac{2^m}{n}).$

Lemma A.5 ^[21] Suppose that $\{X_n; n\geq 1\}$ is a LNQD sequence of random variables with $EX_n = 0.$ Then for any $p > 1$, there exists a positive constant C such that

Acknowledgments

This work is supported by the Natural Science Foundation of Anhui Province of China (Grant No. 1908085QA29) and the Key Natural Science Research Project in Universities of Anhui Province (Grant No. KJ2019A0557).

Conflict of interest

The authors declare no conflict of interest in this paper.