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

For a random field $z(t)$ defined for t ∈ R ⊆ Rᵈ 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.