On fixed-domain asymptotics, parameter estimation and isotropic Gaussian random fields with Matérn covariance functions

Abstract

A method is proposed for estimating the microergodic parameters (including the smoothness parameter) of stationary Gaussian random fields on ${\mathbb{R}}^{\mathit{d}}$ with isotropic Matérn covariance functions using irregularly spaced data. This approach uses higher-order quadratic variations and is applied to three designs, namely stratified sampling design, randomized sampling design and deformed lattice design. Microergodic parameter estimators are constructed for each of the designs. Under mild conditions, these estimators are shown to be consistent with respect to fixed-domain asymptotics. Upper bounds to the convergence rate of the estimators are also established. A simulation study is conducted to gauge the accuracy of the proposed estimators.