This thesis contains three papers focusing on estimation and inference in the spatial data. In the first paper (Chapter 2), we study the estimation and inference in spatial varying coefficient models for data distributed over complex domains. We use bivariate splines over triangulations to represent the coefficient functions. The estimators of the coefficient functions are consistent, and rates of convergence of the proposed estimators are established.

In the second paper (Chapter 3), we extend the idea from the first project and consider a class of flexible partially linear spatially varying coefficient autoregressive models. Under some regularity conditions, the estimated constant coefficients are asymptotically normally distributed, and the estimated varying coefficients are consistent and possess the optimal convergence rate. We further develop an efficient algorithm to calculate the distance between neighbors over complex domains. In addition, we propose a model selection approach to identify explanatory variables with constant and varying effects. The proposed method is much more computationally efficient than the local smoothing method such as the geographically weighted regression, and thus capable of handling large scale of spatial data. The performance of the proposed estimation and model identification methods are evaluated by two simulation examples and the Sydney real estate dataseet.

Note that selecting the smoothing parameter and triangulation mesh is a critical part of im- plementing the bivariate penalized spline fitting in the first paper and second paper. In the third paper (Chapter 4), we propose using the generalized cross-validation to show how to select the roughness penalty parameter and the triangulation jointly. We demonstrate the effectiveness of the proposed method via extensive numerical studies. We extend our method to spatially varying coefficient models, a generalization of linear regression models for exploring non-stationarity of a regression relationship in spatial data analysis.