Berry–Esseen type bounds in heteroscedastic semi-parametric model

Consider the heteroscedastic semi-parametric model y i = x i β + g ( t i ) + σ i e i ( 1 ≤ i ≤ n ) , where σ i 2 = f ( u i ) , 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 { e i } 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 { e i } 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 / 3 ) , O ( n − 1 / 12 + ϖ / 6 ) and O ( n − 1 / 12 + ϖ / 6 ) , respectively, where 0 < ϖ < 1 / 2 .