The Berry–Esseen-type bound for the G-M estimator in a nonparametric regression model with α -mixing errors
Yan Wang, Wei Yu, Xiaoqin Li, Xuejun Wang · Statistics · 2022
Consider the fixed design nonparametric regression model: Yi=g(xi)+εi,i=1,2,…,n,n≥1, where xi are known fixed design points from [0,1] satisfying 0=x0≤x1≤⋯≤xn−1≤xn=1 and g(x) is an unknown function defined on the closed interval [0,1], Yi are the response variables, and εi are zero mean α-mixing random variables with α(n)=O(n−λ)forsomeλ>(2+δ)(1+δ)/δ, where δ>0. We derive the Berry–Esseen-type bound for the G-M estimator of g(⋅) under some mild conditions, which approximates to O(n−14) provided that weights and moments are appropriate. Our results generalize some known ones in the literature. In addition, a simulation study is given to illustrate the validity of the theoretical result.