Simultaneous and post-selection inference for mixed parameters

Abstract

This thesis primarily focuses on the development of statistically valid tools for simultaneous and post-selection inference for a mixed parameter under linear mixed models (LMM) and generalised LMM (GLMM) as well as the investigation of their performance in practice. First, we construct simultaneous confidence intervals for mixed parameters using the max-type statistic, which is readily applicable in the multiple testing procedure. We show that the cluster-wise inference is statistically invalid once we deal with joint statements and it may lead to completely erroneous conclusions. Second, we deal with the simultaneous inference for empirical best predictors in GLMM. Finally, we investigate the issue of post-selection inference for a mixed parameter using conditional Akaike information criterion as a model selection procedure. Within the framework of LMM, we develop a complete theory to construct confidence intervals for mixed parameters under three frameworks: nested and general models, as well as a misspecified setting