Local linear spatial regression

Abstract

A local linear kernel estimator of the regression function x↦g(x):=E[Y_i|X_i=x], x∈ℝ^d, of a stationary (d+1)-dimensional spatial process {(Y _i,X_i),i∈ℤ^N} observed over a rectangular domain of the form ℐ_n:={i=(i₁,…,i_N)∈ℤ^N|1≤i_k≤n_k,k=1,…,N}, n=(n₁,…,n_N)∈ℤ^N, is proposed and investigated. Under mild regularity assumptions, asymptotic normality of the estimators of g(x) and its derivatives is established. Appropriate choices of the bandwidths are proposed. The spatial process is assumed to satisfy some very general mixing conditions, generalizing classical time-series strong mixing concepts. The size of the rectangular domain ℐ_n is allowed to tend to infinity at different rates depending on the direction in ℤ^N.