Random Orthogonal Matrices with Applications in Statistics

Abstract

<p>This dissertation focuses on random orthogonal matrices with applications in statistics. While Bayesian inference for statistical models with orthogonal matrix parameters is a recurring theme, several of the results on random orthogonal matrices may be of interest to those in the broader probability and random matrix theory communities. In Chapter 2, we parametrize the Stiefel and Grassmann manifolds, represented as subsets of orthogonal matrices, in terms of Euclidean parameters using the Cayley transform and then derive Jacobian terms for change of variables formulas. This allows for Markov chain Monte Carlo simulation from probability distributions defined on the Stiefel and Grassmann manifolds. We also establish an asymptotic independent normal approximation for the distribution of the Euclidean parameters corresponding to the uniform distribution on the Stiefel manifold. In Chapter 3, we present polar expansion, a general approach to Monte Carlo simulation from probability distributions on the Stiefel manifold. When combined with modern Markov chain Monte Carlo software, polar expansion allows for routine and flexible posterior inference in models with orthogonal matrix parameters. Chapter 4 addresses prior distributions for structured orthogonal matrices. We introduce an approach to constructing prior distributions for structured orthogonal matrices which leads to tractable posterior simulation via polar expansion. We state two main results which support our approach and offer a new perspective on approximating the entries of random orthogonal matrices.</p>