### MICHAEL L. STEIN

Department of Statistics, The University of Chicago, 5734 University Avenue,
Chicago, IL 60637, U.S.A.

(Received September 6, 1988; revised February 13, 1990)

Abstract.    Suppose a two-dimensional spatial process z(x) with generalized covariance function G(x, x') \propto |x-x'|2 log|x-x'| (Matheron, 1973, Adv. in Appl. Probab., 5, 439-468) is observed with error at a number of locations. This paper gives a kernel approximation to the optimal linear predictor, or kriging predictor, of z(x) under this model as the observations get increasingly dense. The approximation is in terms of a Kelvin function which itself can be easily approximated by series expansions. This generalized covariance function is of particular interest because the predictions it yields are identical to an order 2 thin plate smoothing spline. For moderate sample sizes, the kernel approximation is seen to work very well when the observations are on a square grid and fairly well when the observations come from a uniform random sample.

Key words and phrases:    Thin plate spline, prediction of random fields, Kelvin function, nonparametric regression.

Source ( TeX , DVI , PS )