Fixed design regression for negatively associated random fields

Abstract

Data collected on the surface of the earth at different sites often have two- or three-dimensional coordinates associated with it. We assume a simple setting where these sites are integer lattice points, say, 𝒵 N , N ≥ 1, in the N-dimensional Euclidean space R N . Denote n = (n 1, …, n N )∈𝒵 N and I n = {i: i∈𝒵 N , 1 ≤ i k ≤n k , k = 1, …, N}. Consider a simple regression model where the design points x ni 's and the responses Y ni 's are related as follows: Y ni = g(x ni )+ϵ ni , i∈I n , where x ni 's are fixed design points taking values in a compact subset of R d and where g is a bounded real-valued function defined on R d and ϵ ni are negatively associated random disturbances with zero means and finite variances. The function g(x) is estimated by a general linear smoother g n (x). The asymptotic normality of the estimate g n (x) is established under weak conditions, and general conditions under which the bias g n (x) tends to zero are also determined.