Asymptotic and computational methods in spatial statistics

The dissertation consists of two parts. The first part studies connection between fixed-domain asymptotics and the equivalence of Gaussian measures. It is stressed that one of the most important probabilistic tools to establish fixed-domain asymptotics is to use the equivalence and singularity of Gaussian measures along with related criteria. Two alternative proofs are attempted to show some results about asymptoic optimality of prediction and the equivalence of two Gaussian measures using reproducing kernel Hilbert space, which may have potential power to give preferable conditions for fixed-domain asymptotics in spatial domain without constrains like stationarity of underlying processes.The second part of the dissertation pertains to the application of covariance tapering to deal with large spatial data sets. When the spatial sample size is extremely large, as for many environmental and ecological studies, operations on the large covariance matrix are numerically challenging. Covariance tapering is a technique to alleviate these numerical challenges. We investigate how tapering affects the asymptotic efficiency of the maximum likelihood estimator (MLE) and establish asymptotic properties, particularly asymptotic distributions of the exact MLEs and tapered MLEs under the fixed-domain asymptotic framework for the Matern model. We show that under some conditions on the tapering function, the tapered MLE is asymptotically as efficient as the true MLE for the microergodic parameter in the Matern model. For the general setting, we compare the exact and tapered likelihood and their derivatives in seeking conditions on tapering which may yield no loss of efficiency. The convergence rate of effect of tapering on prediction is also studied. Finally, The computational gain and comparable estimation are illustrated by simulation studies and an application to the US precipitation data for April 1948.