Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure

Abstract

For a random field z(t) defined for $t \in R \subseteq \mathbb{R}^d$ with specified second-order structure (mean function m and covariance function K), optimal linear prediction based on a finite number of observations is a straightforward procedure. Suppose (m0, K0) is the second-order structure used to produce the predictions when in fact (m1, K1) is the correct second-order structure and (m0, K0) and (m1, K1) are "compatible" on R. For bounded R, as the points of observation become increasingly dense in R, predictions based on (m0, K0) are shown to be uniformly asymptotically optimal relative to the predictions based on the correct (m1, K1). Explicit bounds on this rate of convergence are obtained in some special cases in which K0 = K1. A necessary and sufficient condition for the consistency of best linear unbiased predictors is obtained, and the asymptotic optimality of these predictors is demonstrated under a compatibility condition on the mean structure.