Methodological Advances for Multi-group Data

Abstract

This dissertation focuses on improving inference in analyses of multi-group data, that is, data obtained from non-overlapping subpopulations such as across counties in a state or for various socio-economic groups. Precise and accurate group-specific inference based on such data may be encumbered by small within-group sample sizes. In such cases, inference may be improved by making use of auxiliary information. In this work, we present two streams of methodological development aimed at improving group-specific inference for multi-group data that may feature small within-group sample sizes for some or all of the groups. First, we detail methodology that constructs frequentist-valid prediction regions based on indirect information. We show such prediction regions may feature improved precision over those constructed with standard approaches. To this end, we present methods that result in accurate and precise prediction regions for multi-group data based on a continuous response in Chapter 2 and a categorical response in Chapter 3. We develop straightforward computational algorithms to compute the regions and detail empirical Bayesian estimation procedures that allow for information to be shared across groups in the construction of the prediction regions. In Chapter 4, we present work that improves covariance estimation for structured multi-group data with shrinkage estimation that allows for robustness to structural assumptions. In particular, for multi-group matrix-variate data, we describe a hierarchical prior distribution that improves covariance estimate accuracy by flexibly allowing for shrinkage within groups towards a Kronecker structure and across groups towards a pooled covariance estimate. We illustrate the utility of all methods presented with simulation studies and data applications.