Uniformly asymptotic normality of the weighted estimator in nonparametric regression model with φ-mixing errors

Abstract

Consider the following nonparametric model: Y ni =g(x ni )+ε ni , 1 ⩽ i ⩽ n, where x n i ∈ A $ x_{ni}\in \mathbb{A} $ are the nonrandom design points and A $ \mathbb{A} $ is a compact set of ℝ m for some m=1, g (·) is a real valued function defined on A $ \mathbb{A} $ , and ε n1, ⋅, ε nn are zero mean φ-mixing random errors with finite moment of 2+δ order for some δ>0. Under some general conditions, we obtain the uniformly asymptotic normality of the weighted estimator of g (·). The rate of Berry-Esseen bound can approximate O(n −1/4) under some appropriate conditions. The validity of the main results is partially illustrated via a numerical simulation.