Bayesian Estimation of the Shape Parameters of Mcdonald Generalized Beta-Binomial Distribution

The paper explores and establishes a unique Bayesian framework for estimating three shape parameters of the McDonald generalized beta-binomial distribution. The mixture distribution is used in modelling overdispersed binomial data. Foundations of the framework have been enriched by knowledge of Bayesian statistics and Markov Chain Monte Carlo methods. A Metropolis within Gibbs Monte Carlo method to sample from the unknown posterior form of the distribution was used. The shape parameters (α, β and γ) were assigned flat gamma priors to ensure equal probabilities for all the values. McDonald generalized beta-binomial variables were simulated with fixed shape parameters set at (α,β,γ)=(0.5,0.5,0.5) respectively and samples generated were used to estimate the parameters, to evaluate if the method recovers estimates close to the true parameter values. Standard errors were also computed for the simulated data and real data. Further, credible regions and highest probability density intervals (HPD) were computed and their corresponding lengths. To evaluate the marginal posterior samples for every shape parameter generated trace plots presented, their respective correlation plots were also presented and the histograms to show the distributions assumed by every parameter. Bayesian framework provides a direct and flexible method of computation for a mixture distribution whose complexity may pose challenges of integration when using the classical methods of estimation.