Fixed design regression for negatively associated random fields

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, 𝒵N, N ≥ 1, in the N-dimensional Euclidean space RN. Denote n = (n1, …, nN)∈𝒵N and In = {i: i∈𝒵N, 1 ≤ ik≤nk, k = 1, …, N}. Consider a simple regression model where the design points xni's and the responses Yni's are related as follows: Yni = g(xni)+ϵni, i∈In, where xni's are fixed design points taking values in a compact subset of Rd and where g is a bounded real-valued function defined on Rd and ϵni are negatively associated random disturbances with zero means and finite variances. The function g(x) is estimated by a general linear smoother gn(x). The asymptotic normality of the estimate gn(x) is established under weak conditions, and general conditions under which the bias gn(x) tends to zero are also determined.