Berry-Esseen type bounds in heteroscedastic semi-parametric model

摘要

Consider the heteroscedastic semi-parametric model yi=xiΒ+g(t_i)+σ_ie_i (1≤i≤n), where σ_i~2=f(u_i), the design points (x_i,t_i,u_i) 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<π{variant}<1/2.