This thesis has two themes. One is missing data analysis, and the other is spatial data analysis.

Missing data frequently occur in many statistics problems. It can arise naturally in many applications. For example, in many surveys there are data that could have been observed are missing due to non-response. It can also be a deliberate modeling choice. For example, a mixed effects model can include random variables that are not observable (called latent variables or random effects). Imputation is often used to facilitate parameter estimation in the presence of

missing data, which allows one to use the complete sample estimators on the imputed data set. Parametric fractional imputation (PFI) is an imputation method proposed by Kim (2011), which simplifies the computation associated with the EM algorithm for maximum likelihood estimation with missing data. In this thesis we study four extensions of the PFI methods: 1. The use of PFI to handle non-ignorable non-response problem in linear and generalized linear mixed models. 2. Application of PFI method for quantile estimation with missing data. 3. Likelihood-based inference for missing data using PFI. 4. A semiparametric fractional imputation method for handling missing covariate.

The second theme is spatial data analysis. Estimation of the covariance structure of spatial processes is of fundamental importance in spatial statistics. The difficulty arises when spatial process exhibits non-stationarity or the observed spatial data is irregularly spaced. We propose estimation methods targeting to solve these two difficulties. 1. We propose a non-stationary spatial modeling, study the theoretical properties of estimation and plug-in kriging prediction of a non-stationary spatial process, and explore the connection between kriging under non-stationary models and spatially adaptive non-parametric smoothing methods. 2. A semiparametric estimation of spectral density function for irregular spatial data.