LINEAR MIXED MODEL ESTIMATION WITH DIRICHLET PROCESS RANDOM EFFECTS By CHEN LI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LINEAR MIXED MODEL ESTIMATION WITH DIRICHLET PROCESS RANDOM EFFECTS By Chen Li August 2012 Chair: George Casella Major: Statistics The linear mixed model is very popular, and has proven useful in many areas of applications. (See, for example, McCulloch and Searle (2001), Demidenko (2004), and Jiang (2007).) Usually people assume that the random effect is normally distributed. However, as this distribution is not observable, it is possible that the distribution of the random effect is non-normal (Burr and Doss (2005), Gill and Casella (2009), Kyung et al. (2009, 2010)). We assume that the random effect follows a Dirichlet process, as discussed in Burr and Doss (2005), Gill and Casella (2009), Kyung et al. (2009, 2010). In this dissertation, we first consider the Dirichlet process as a model for classical random effects, and investigate their effect on frequentist estimation in the linear mixed model. We discuss the relationship between the BLUE (Best Linear Unbiased Estimator) and OLS (Ordinary Least Squares) in Dirichlet process mixed models, and also give conditions under which the BLUE coincides with the OLS estimator in the Dirichlet process mixed model. In addition, we investigate the model from the Bayesian view, and discuss the properties of estimators under different model assumptions, compare the estimators under the frequentist model and different Bayesian models, and investigate minimaxity. Furthermore, we apply the linear mixed model with Dirichlet Process random effects to a real data set and get satisfactory results.