## The Annals of Statistics

### Asymptotic Distributions of Minimum Norm Quadratic Estimators of the Covariance Function of a Gaussian Random Field

Michael Stein

#### 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.

#### Article information

Source
Ann. Statist., Volume 17, Number 3 (1989), 980-1000.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347252

Digital Object Identifier
doi:10.1214/aos/1176347252

Mathematical Reviews number (MathSciNet)
MR1015134

Zentralblatt MATH identifier
0681.62027

JSTOR