Fixed Domain Asymptotics and Consistent Estimation for Gaussian Random Field Models in Spatial Statistics and Computer Experiments

Abstract

Gaussian random fields are commonly used as models for spatial processes. This study establishes consistent parameter estimation of Gaussian random field models in computer experiments under fixed domain asymptotics. It has two parts. The first part deals with isotropic covariance function. Maximum likelihood is a preferred method for estimating the covariance parameters. However, when the sample size n is large, it is a burden to compute the likelihood. Covariance tapering is an effective technique to approximate the covariance function with a taper (usually a compactly supported correlation function) so that the computation can be reduced. This part studies the fixed domain asymptotic behavior of the tapered MLE for the microergodic parameter of isotropic Matern class covariance function when the taper support is allowed to shrink as n tends to infinity. In particular when d<4, conditions are established in which the tapered MLE is strongly consistent and asymptotically normal.

The second part establishes consistent estimators of the covariance and scale parameters of Gaussian random field with multiplicative covariance function. When d = 1, in some cases it is impossible to consistently estimate them simultaneously under fixed domain asymptotics. However, when d > 1, consistent estimators of functions of covariance and scale parameters can be constructed by using quadratic variation and spectral analysis. Consequently, they provide the consistent estimators of the covariance and scale parameters.