UC Riverside

Several bivariate beta distributions have been proposed in the literature. In

particular, Olkin and Liu (2003) proposed a 3 parameter bivariate beta model, which Arnold and Ng (2011) extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this dissertation is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach.

Chapter one briely describes the univariate beta distribution and its properties. The 5 and 8 parameter bivariate beta distribution is defined and estimation strategies are discussed. Chapter two is dedicated to the background of approximate Bayesian computation (ABC), where the foundation and groundwork is laid. Toy examples are provided to better understand the algorithm and to study its properties. Chapter three is the application of ABC to the 5 and 8 parameter bivariate beta model. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors, sample sizes, and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed. Chapter four attempts to lay the ground work to modify existing ABC (accept reject) algorithms to search for maximum likelihood type estimates in the absence of the likelihood function. Examples are provided to demonstrate the relationship between maximum likelihood estimation and acceptance rates. Algorithms are proposed and applied to data sets in an attempt to search for maximum likelihood type estimates using only sufficient statistics. Results are compared to the known maximum likelihood estimates.