Approximate Bayesian Computation for a Five-Parameter Bivariate Beta Model and Its Applications

• Activity Number: 401
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Computing
• Author(s): Roberto Crackel*+ and James Flegal
• Companies: and University of California, Riverside
• Keywords: Approximate Bayesian computation ; Bivariate beta ; accept-reject

Abstract

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 a 5 and 8 parameter model. 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 research is on Bayesian estimation for the 5 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried under various priors and tolerance levels. We make application of the 5 parameter bivariate beta in a bivariate beta binomial context. Specifically, the bivariate beta binomial distribution is used to model the probabilities and correlation of purchasing bacon and eggs on a single shopping trip, described previously by Danaher and Hardie (2005). In this model, we allow the 5 parameter model to serve as a prior distribution to correlated proportions and it serves as a competitor to the bivariate beta model prop