Consistency properties for the wavelet estimator in nonparametric regression model with dependent errors

Qihui He,Mingming Chen

doi:10.1186/s13660-021-02603-0

Abstract

In this paper, we establish the pth mean consistency, complete consistency, and the rate of complete consistency for the wavelet estimator in a nonparametric regression model with m-extended negatively dependent random errors. We show that the best rates can be nearly O(n^{-1/3}) under some general conditions. The results obtained in the paper markedly improve and extend some corresponding ones to a much more general setting.

Highlights

Consider the nonparametric regression modelYni = g(tni) + εni, 1 ≤ i ≤ n, n ≥ 1, (1)where the regression function g is an unknown Borel-measurable function defined on [0, 1], {tni} are nonrandom design points such that 0 ≤ tn1 ≤ · · · ≤ tnn ≤ 1, and {εni} are random errors.It is known that nonparametric regression model (1) has many applications in practical fields such as communications and control systems, classification, econometrics, and so on
Yang and Wang [6] investigated the strong consistency of the P-C estimator in (1) with negatively associated (NA) samples; Yang [7] studied the rate of asymptotic normality for the weighted estimator with NA samples; Liang and Jing [8] established the mean consistency, strong consistency, complete consistency, and asymptotic normality for the weighted estimator with NA samples; Wang et al [9] investigated
We further investigate the consistency properties of estimator (2) in the nonparametric regression model (1) based on m-extended negatively dependent (END) random errors

Summary

Consider the nonparametric regression model

Where the regression function g is an unknown Borel-measurable function defined on [0, 1], {tni} are nonrandom design points such that 0 ≤ tn1 ≤ · · · ≤ tnn ≤ 1, and {εni} are random errors. Yang and Wang [6] investigated the strong consistency of the P-C estimator in (1) with negatively associated (NA) samples; Yang [7] studied the rate of asymptotic normality for the weighted estimator with NA samples; Liang and Jing [8] established the mean consistency, strong consistency, complete consistency, and asymptotic normality for the weighted estimator with NA samples; Wang et al [9] investigated He and Chen Journal of Inequalities and Applications (2021) 2021:152 the complete consistency of the weighted estimator in (1) with extended negatively dependent (END) errors; Chen et al [10] obtained the mean consistency, strong consistency, and complete consistency of the weighted estimator in model (1) with martingale difference errors. Definition 1.2 A function space Hν (ν ∈ R) is called the Sobolev space of order ν if for all h ∈ Hν , we have |h (ω)|2(1 + ω2)ν dω < ∞, where his the Fourier transform of h

Define the wavelet kernel

Hence we only need to prove n p

Ani n

Ani and