On fixed-domain asymptotics and covariance tapering in Gaussian random field models

Abstract

Gaussian random fields are commonly used as models for spatial processes and maximum likelihood is a preferred method of choice for estimating the covariance parameters. However if the sample size n is large, evaluating the likelihood can be a numerical challenge. Covariance tapering is a way of approximating the covariance function with a taper (usually a compactly supported function) so that the computational burden is reduced. This article studies the fixed-domain asymptotic behavior of the tapered MLE for the microergodic parameter of a Matérn covariance function when the taper support is allowed to shrink as n→α. In particular if the dimension of the underlying space is ≤ 3, conditions are established in which the tapered MLE is strongly consistent and also asymptotically normal. Numerical experiments are reported that gauge the quality of these approximations for finite n.