Estimating the parameters of some common Gaussian random fields with nugget under fixed-domain asymptotics

This article considers parameter estimation for a class of Gaussian random fields on [0,1)d that are observed with measurement error and irregularly spaced design sites. This class comprises Gaussian random fields with suitably smooth mean functions and isotropic powered exponential, Matern or generalized Wendland covariance functions. Under fixed-domain asymptotics, consistent estimators are proposed for three microergodic parameters, namely the nugget, the smoothness parameter and a parameter related to the coefficient of the principal irregular term of the isotropic covariance function. Upper bounds for the convergence rate of these estimators are established. Simulations are conducted to study the finite sample accuracy of the proposed estimators.