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June, 1990 Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure
Michael Stein
Ann. Statist. 18(2): 850-872 (June, 1990). DOI: 10.1214/aos/1176347629

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.

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Michael Stein. "Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure." Ann. Statist. 18 (2) 850 - 872, June, 1990. https://doi.org/10.1214/aos/1176347629

Information

Published: June, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0716.62099
MathSciNet: MR1056340
Digital Object Identifier: 10.1214/aos/1176347629

Subjects:
Primary: 62M20
Secondary: 41A25, 60G60

Rights: Copyright © 1990 Institute of Mathematical Statistics

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Vol.18 • No. 2 • June, 1990
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