Berry–Esseen type bounds of estimators in a semiparametric model with linear process errors

Consider the semiparametric regression model yi=xiβ+g(ti)+Vi,1≤i≤nyi=xiβ+g(ti)+Vi,1≤i≤n, where ββ is an unknown parameter of interest, (xi,ti)(xi,ti) are nonrandom design points, yiyi are the response variables, g(⋅)g(⋅) is an unknown function defined on the closed interval [0,1][0,1], and the correlated errors Vi=∑j=−∞∞ψjei−j, with ∑j=−∞∞|ψj|<∞, and eiei are negatively associated random variables. Under appropriate conditions, in this paper, we derive Berry–Esseen type bounds for estimators of ββ and g(⋅)g(⋅). As a corollary, by making a certain choice of the weights, we give the Berry–Esseen type bounds for estimators of ββ and g(⋅)g(⋅); they are O(n−1/4(logn)3/4)O(n−1/4(logn)3/4) and O(n−3/28(logn)9/28)O(n−3/28(logn)9/28), respectively, and under further restriction for the weights, the Berry–Esseen type bound for estimator of g(⋅)g(⋅) can also attain O(n−1/4(logn)3/4)O(n−1/4(logn)3/4).