Asymptotic Distributions of Minimum Norm Quadratic Estimators of the Covariance Function of a Gaussian Random Field
The Annals of Statistics · 1989 · Volume 17 · Issue 3
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
#39;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#39;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.Keywords
60G30, 60G60, 62M20, Equivalence of Gaussian measures, Geostatistics, kriging
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