Inference on the double binomial distribution

The double binomial distribution was first introduced by Efron (1986) as a probability model for clustered binary data. As it’s well known, the dependence structure between such data points often exceed that predicted by the mean-variance relation, rendering it misleading to analyze such data with the binomial distribution. The two parameter double binomial distribution was introduced by Efron to serve as an alternative to the binomial distribution that accommodates the additional variability in clustered binary data. In this article, we provide approaches for obtaining maximum likelihood and Bayesian estimators of the parameters of the double binomial distribution. We derive Jeffrey’s priors for these parameters, and show that these priors facilitate the derivation of fully conditional posterior distributions thus ensuring that a Gibbs sampling procedure is used to sample from the posterior distributions of the parameters. We will illustrate our work with simulated and real data sets.