Topics in Multivariate Statistics with Dependent Data

Abstract

This dissertation comprises four chapters. The first is an introduction to the topics of the dissertation and the remaining chapters contain the main results. Chapter 2 gives new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. The presented theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full data. The theory is applied to two multivariate mixed models for which it was unknown whether maximum likelihood estimators are consistent. In Chapter 3 an algorithm is proposed for maximum likelihood estimation of a covariance matrix when the corresponding correlation matrix can be written as the Kronecker product of two lower-dimensional correlation matrices. The proposed method is fully likelihood-based. Some desirable properties of separable correlation in comparison to separable covariance are also discussed. Chapter 4 is concerned with Bayesian vector autoregressions (VARs). A collapsed Gibbs sampler is proposed for Bayesian VARs with predictors and the convergence properties of the algorithm are studied. The Markov chain generated by the algorithm is proved to be geometrically ergodic, regardless of whether the number of observations in the VAR is small or large in comparison to the order and dimension of the VAR. It is also established that the geometric convergence rate is bounded away from one as the number of observations tends to infinity.