Bayesian inference for two nonstandard flexible families of bivariate Kumaraswamy models: theory and applications

Arnold and Ghosh (2017a, 2017b) have proposed a broad spectrum of bivariate Kumaraswamy (henceforth, denoted by KW) distributions involving conditional specification, conditional survival specification, and starting from the Arnold and Ng (2011) eight parameter bivariate beta model. In addition, copula based construction of bivariate KW models was considered. Included among the models that they discussed were the Olkin-Trikalinos (denoted by OT-BK) and the Ghosh-BK (denoted by G-BK) models. These two models can accommodate both positive and negative correlation under certain parametric restrictions. However, this comes at the expense of dealing with a density that is mathematically intractable. We focus our attention on estimation in the 4 and 5 parameter OT-BK and the G-BK models, respectively, using a Bayesian approach. A general framework based on approximate Bayesian computation methodology is proposed and studied in this article. In particular, the choice of priors must be such that they satisfy the parameter constraints for these models. We conduct simulation studies for both of these models under a wide selection of priors. For illustrative purposes, a real data set has been re-analyzed.