Kernel regression estimation for continuous spatial processes

Abstract

We investigate here a kernel estimate of the spatial regression function r(x) = E(Y u | X u = x), x ∈ ℝd, of a stationary multidimensional spatial process { Z u = (X u, Y u), u ∈ ℝ N }. The weak and strong consistency of the estimate is shown under sufficient conditions on the mixing coefficients and the bandwidth, when the process is observed over a rectangular domain of ℝN. Special attention is paid to achieve optimal and suroptimal strong rates of convergence. It is also shown that this suroptimal rate is preserved by using a suitable spatial sampling scheme.