## The Annals of Statistics

- Ann. Statist.
- Volume 18, Number 2 (1990), 850-872.

### Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 18, Number 2 (1990), 850-872.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176347629

**Digital Object Identifier**

doi:10.1214/aos/1176347629

**Mathematical Reviews number (MathSciNet)**

MR1056340

**Zentralblatt MATH identifier**

0716.62099

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 41A25: Rate of convergence, degree of approximation 60G60: Random fields

**Keywords**

Kriging spatial statistics approximation in Hilbert spaces

#### Citation

Stein, Michael. Uniform Asymptotic Optimality of Linear Predictions of a Random Field Using an Incorrect Second-Order Structure. Ann. Statist. 18 (1990), no. 2, 850--872. doi:10.1214/aos/1176347629. https://projecteuclid.org/euclid.aos/1176347629