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

Abstract

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