Berry–Esseen type bounds in heteroscedastic semi-parametric model

Consider the heteroscedastic semi-parametric model yi=xiβ+g(ti)+σieiyi=xiβ+g(ti)+σiei(1≤i≤n)(1≤i≤n), where σi2=f(ui), the design points (xi,ti,ui) are known and nonrandom, the functions g(·)g(·) and f(·)f(·) are defined on closed interval [0,1]. When the random errors {ei}{ei} are assumed to be a sequence of stationary α‐mixingα‐mixing random variables, we derive the Berry–Esseen type bounds for the estimators of ββ and g(·)g(·) under f(·)f(·) is known, respectively. When f(·)f(·) is unknown, the Berry–Esseen type bounds for the estimators of ββ, g(·)g(·) and f(·)f(·) are discussed under the errors {ei}{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 ββ, g(·)g(·) and f(·)f(·) can achieve O(n−1/6+ϖ/3)O(n−1/6+ϖ/3), O(n−1/12+ϖ/6)O(n−1/12+ϖ/6) and O(n−1/12+ϖ/6)O(n−1/12+ϖ/6), respectively, where 0<ϖ<1/20<ϖ<1/2.