Kernel regression estimation for random fields

Consider a stationary random field {Xn}{Xn} indexed by NN-dimensional lattice points, where {Xn}{Xn} takes values in RdRd. An important problem in spatial statistics is the estimation of the regression of {Xn}{Xn} on the values of the random field at surrounding sites, say, Xn1,…,XnℓXn1,…,Xnℓ. Note that (Xn1,…,Xnℓ)(Xn1,…,Xnℓ) is a ℓdℓd-dimensional vector. Assume the existence of the regression function r(x)=E{ϕ(Xn)|(Xn1,…,Xnℓ)=x},r(x)=E{ϕ(Xn)|(Xn1,…,Xnℓ)=x},where ϕϕ is a continuous real-valued function which is not necessarily bounded, and x∈Rℓdx∈Rℓd. Kernel-type estimators of the regression function r(x)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∞L∞. The results hold for a large class of spatial processes.