This dissertation explores parameter estimation for Gaussian random fields and multivariate Gaussian random processes under fixed-domain asymptotics, a crucial framework for modeling spatial and temporal data. Unlike increasing-domain asymptotics, fixed-domain asymptotics involve a growing number of observations within a fixed, bounded region, leading to denser data. This scenario is common in applications such as image processing, where the spatial domain is constrained by the finite size of the sensor array. First, we study the parameter estimation for a Gaussian field with a multiplicative covariance function, which is particularly relevant in computer experiments. We propose an increment-based estimator for estimating variance and scale parameters, and subsequent analysis shows that the estimator is both strongly consistent and asymptotically normal. Next, we extend the analysis to the bivariate Ornstein-Uhlenbeck process, constructing an explicit estimator that is strongly consistent and asymptotically normal. This estimator, requiring no prior parameter information, is shown to have the same asymptotic covariance matrix as that of the maximum likelihood estimator (MLE).Finally, we investigate asymptotic properties of MLE for the isotropic powered exponential field. Unlike the Mat\'ern model, the spectral density of the powered exponential model poses analytical challenges. We also establish conditions for the equivalence of Gaussian measures, providing a contrast to the orthogonality conditions found in earlier studies.
Read
