Kernel regression estimation for random fields

Consider a stationary random field { X_n } indexed by N-dimensional lattice points, where { X_n } takes values in R^d. An important problem in spatial statistics is the estimation of the regression of { X_n } on the values of the random field at surrounding sites, say, X_n1, ..., X_nℓ. Note that (X_n1, ..., X_nℓ) is a ℓ d-dimensional vector. Assume the existence of the regression function r (x) = E { φ{symbol} (X_n) | (X_n1, ..., X_nℓ) = x },where φ{symbol} is a continuous real-valued function which is not necessarily bounded, and x ∈ R^{ℓ d}. Kernel-type estimators of the regression function r (x) are investigated. They are shown to converge uniformly on compact sets under general conditions. In addition, they can attain the optimal rates of convergence in L_∞. The results hold for a large class of spatial processes.