Berry-Esseen type bounds in heteroscedastic semi-parametric model

Abstract

Consider the heteroscedastic semi-parametric model yi=xiΒ+g(ti)+σiei (1≤i≤n), where σi2=f(ui), the design points (xi,ti,ui) are known and nonrandom, the functions g(·) and f(·) are defined on closed interval [0,1]. When the random errors {ei} are assumed to be a sequence of stationary α-mixing random variables, we derive the Berry-Esseen type bounds for the estimators of Β and g(·) under f(·) is known, respectively. When f(·) is unknown, the Berry-Esseen type bounds for the estimators of Β, g(·) and f(·) are discussed under the errors {ei} are assumed to be independent but not necessarily identically distributed. As corollary, by choosing suitable weighted functions, the Berry-Esseen type bounds for the estimators of Β, g(·) and f(·) can achieve O(n-1/6+π{variant}/3), O(n-1/12+π{variant}/6) and O(n-1/12+π{variant}/6), respectively, where 0 <1/2. © 2011 Elsevier B.V.