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 $(m_0, K_0)$ is the second-order structure used to produce the predictions when in fact $(m_1, K_1)$ is the correct second-order structure and $(m_0, K_0)$ and $(m_1, K_1)$ are "compatible" on $R$. For bounded $R$, as the points of observation become increasingly dense in $R$, predictions based on $(m_0, K_0)$ are shown to be uniformly asymptotically optimal relative to the predictions based on the correct $(m_1, K_1)$. Explicit bounds on this rate of convergence are obtained in some special cases in which $K_0 = K_1$. 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.