| This thesis presents work on two distinct topics: multiple imputation and shrinkage estimation. For the former topic, we provide a general theory on the model uncongeniality issue in multiple imputation and study the relevant hypothesis testing problem; for the latter, we develop a theory of SURE shrinkage estimation in heteroscedastic hierarchical models.;In Chapter 1 of this thesis, we revisit the model uncongeniality issue (Meng, 1994) of multiple imputation inference in light of a series of theoretical results discovered through our study. We first show how the multiple imputation estimator can be treated as a matrix-weighted combination of the imputer's estimator and the user's estimator, reflecting the mixing of information from both parties. We then identify circumstances under which the procedure produces confidence intervals that have at least the declared nominal rate, along with extensions to be used when the original procedure is suspected to fail.;The hypothesis testing problem in multiple imputation is studied in Chapter 2. We first present some modifications of existing Wald-type procedures and then propose several procedures that directly combine the p-values. We illustrate the performance of our method through simulated examples.;Chapter 3 and 4 develop in-depth a theory of SURE shrinkage estimation in heteroscedastic hierarchical models. In chapter 3, we focus on the classic normal models: two classes of SURE shrinkage estimators, the parametric and semi-parametric ones, are introduced and their optimality properties are carefully studied. The method is applied to several real examples and encouraging results are observed. A generalization of the idea to hierarchical models based on natural exponential family with quadratic variance function is then presented in Chapter 4.;It is emphasized that the topics discussed here all have a Bayesian motivation but the studies focus on their frequentist properties, reflecting our firm belief that sound statistical methods should naturally arise from the interplay between Bayesian and frequentist reasoning. |
