## The Annals of Statistics

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Secondary: 60G30: Continuity and singularity of induced measures 60G60: Random fields

**Keywords**

Equivalence of Gaussian measures kriging geostatistics

#### Citation

Stein, Michael. Asymptotic Distributions of Minimum Norm Quadratic Estimators of the Covariance Function of a Gaussian Random Field. Ann. Statist. 17 (1989), no. 3, 980--1000. doi:10.1214/aos/1176347252. https://projecteuclid.org/euclid.aos/1176347252