The consistency for estimator of nonparametric regression model based on NOD errors

Abstract

By using some inequalities for NOD random variables, we give its application to investigate the nonparametric regression model based on these errors. Some consistency results for the estimator of g(x) are presented, including the mean convergence, uniform convergence, almost sure convergence and convergence rate. We generalize some related results and as an example of designed assumptions for weight functions, we give the nearest neighbor weights.

Bernstein-Type Inequality for Widely Dependent Sequence and Its Application to Nonparametric Regression Models

“…Liang and Jing [11] presented some asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences. Yang et al [12] generalized the results of Liang and Jing [11] for negatively associated sequences to the case of negatively orthant dependent sequences and obtained the strong consistency for the estimator of the nonparametric regression models based on negatively orthant dependent errors. Wang et al [13] studied the complete consistency of the estimator of nonparametric regression models based oñ -mixing sequences, and so forth.…”

Weighted sums of strongly mixing random variables with an application to nonparametric regression

“…Consider the following observations Y ni = g(x ni ) + X ni , 1 ≤ i ≤ n, where g is an unknown real-valued regression function that is bounded on A, x ni are known fixed design points from A, {X ni } is a sequence random variables representing the random errors. As in [9,16], we assume that for each n, there is…”

The consistency of estimator under fixed design regression model with NQD errors

“…Hu et al [8] gave the mean consistency, complete consistency, and asymptotic normality of regression models based on linear process errors. Under negatively associated sequences, Liang and Jing [9] presented some asymptotic properties for estimates of nonparametric regression models, Yang et al [10] generalized part results of Liang and Jing [9] for negatively associated sequences to the case of negatively orthant dependent sequences, and so on.…”

“…Liang et al[9]and Yang et al[10] to the case of NQD errors, respectively. And as a consequence, one may get consistency property for the weighted kernel estimators in the model (2.6).Corollary 3.1.…”