Kernel regression estimation for continuous spatial processes

We investigate here a kernel estimate of the spatial regression function r(x) = E(YuXu = x), x ∈ ℝd, of a stationary multidimensional spatial process { Zu = (Xu, Yu), u ∈ ℝN}. 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. © 2007 Allerton Press, Inc.