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September, 1989 Asymptotic Distributions of Minimum Norm Quadratic Estimators of the Covariance Function of a Gaussian Random Field
Michael Stein
Ann. Statist. 17(3): 980-1000 (September, 1989). DOI: 10.1214/aos/1176347252

Abstract

Consider a continuous Gaussian random field $z(x)$ defined on a compact set $R \subset \mathbb{R}^d$ with covariance function of the form $\operatorname{cov}(z(x), z(x')) = \sum^k_{i = 1}\theta_iK_i(x,x')$, where the $K_i$'s are specified and $\theta = (\theta_1, \ldots, \theta_k)'$ is to be estimated. Let $\{x_l\}^\infty_{l = 1}$ be a sequence of distinct points in $R$. Based on $z(x_1), \ldots, z(x_N)$, minimum norm quadratic estimation can be used to estimate $\theta$. Suppose $K_1, \ldots, K_k$ are compatible covariance functions on $R$, which means that the Gaussian measures with means zero and covariance functions $K_1, \ldots, K_k$ are mutually absolutely continuous. Then, as the number of observations $N$ increases, the minimum norm quadratic estimator of $\sum^k_{i = 1}\theta_i$ is asymptotically normal with variance of order $N^{-1}$. The minimum norm quadratic estimator of any other linear combination of the $\theta_i$'s converges (in $L^2$) to some nondegenerate random variable. This limit is the same for any two dense sequence of points in $R$. Thus, a definition of a minimum norm quadratic estimator of $\theta$ when $z(\cdot)$ is observed everywhere in $R$ is obtained.

Citation

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Michael Stein. "Asymptotic Distributions of Minimum Norm Quadratic Estimators of the Covariance Function of a Gaussian Random Field." Ann. Statist. 17 (3) 980 - 1000, September, 1989. https://doi.org/10.1214/aos/1176347252

Information

Published: September, 1989
First available in Project Euclid: 12 April 2007

zbMATH: 0681.62027
MathSciNet: MR1015134
Digital Object Identifier: 10.1214/aos/1176347252

Subjects:
Primary: 62M20
Secondary: 60G30, 60G60

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 3 • September, 1989
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