The consistency for the estimator of nonparametric regression model based on martingale difference errors

Abstract

In this paper, by using the inequalities for martingale difference sequence, we investigate the consistency for the estimator of nonparametric regression model based on martingale difference errors. Some results on consistency for the estimator of g(x) are presented, including the mean consistency, complete consistency and strong consistency. As an application, the consistency for the nearest neighbor estimator is obtained.

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

“…Due to its significant applications, this nonparametric regression model has also been investigated in various dependent cases. For example, 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 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.…”