Parametric Estimation in Spatial Regression Models

Abstract

This dissertation addresses the asymptotic theory behind parametric estimation inspatial regression models. In spatial statistics, there are two prominent types of asymptotic frameworks: increasing domain asymptotics and infill asymptotics. The former assumes that spatial data are observed over a region that increases with the sample size, whereas the latter assumes the observations become increasingly dense in a bounded domain. It is well understood that both frameworks lead to drastically different behavior of classical statistical estimators. Under increasing domain asymptotics, we use recently established limit theorems for random fields to prove consistency and asymptotic normality of estimators in a nonlinear regression model. The theory presented here hinges on a crucial assumption that the covariates and error are independent of one another. However, when covariates also exhibit spatial variation, this assumption of independence becomes questionable. This possibility of spatial correlation between the covariates and error is known as spatial confounding. We examine several possible parametric models of spatial confounding and under increasing domain asymptotics, we determine that the degree of confounding can be estimated with good precision through maximum likelihood methods. Finally, under infill asymptotics, we focus our attention on linear regression models in a Gaussian setting. Existing literature in infill asymptotics tends to ignore estimation of the mean and emphasizes estimation of variance components in the error. For estimation of the mean, the sample path properties of the mean relative to the error play an important role. We show that under certain roughness conditions on the sample paths of the covariates, it is possible to obtain consistent, asymptotically normal estimates of regression parameters through maximum likelihood estimation.