Gaussian random fields are widely studied in various subject areas. This dissertation focuses on estimating covariance parameters of stationary Gaussian random fields based on both regularly and irregularly spaced sampling points, as well as investigating the infill asymptotic properties of the estimators.We first consider a bivariate Gaussian random process and propose an increment-based estimator for the smoothness parameter in the cross-covariance function, for which the strong consistency and asymptotic normality hold under the infill asymptotic framework. We further study the joint asymptotic distribution of estimators for smoothness parameters in the cross-covariance and autocovariance functions. Subsequently, we estimate the scale parameter and range parameters of a univariate anisotropic Ornstein-Uhlenbeck field based on quadratic forms of vectors of observations. The estimators we propose are computationally more efficient than the maximum likelihood estimators but have similar infill asymptotic performances with MLEs. Another computational complexity reduction method we use is the Vecchia approximation. We estimate the scale parameter in the Matérn covariance function using the maximizer of the likelihood approximated by the standard Vecchia approach. We study the bias resulting from a misspecified range parameter and the conditioning variables of the Vecchia approximation. The theoretical results in this work are illustrated by simulations.
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