A generalization of the Binomial distribution based on the dependence ratio

We propose a generalization of the Binomial distribution, called DR‐Binomial, which accommodates dependence among units through a model based on the dependence ratio (Ekholm et al., Biometrika, 82, 1995, 847). Properties of the DR‐Binomial are discussed, and the constraints on its parameter space are studied in detail. Likelihood‐based inference is presented, using both the joint and profile likelihoods; the usefulness of the DR‐Binomial in applications is illustrated on a real dataset displaying negative unit‐dependence, and hence under‐dispersion compared with the Binomial. Although the DR‐Binomial turns out to be a reparameterization of Altham's Additive‐Binomial and Kupper–Haseman's Correlated‐Binomial distribution, we believe its introduction is useful, both in terms of interpretability and mathematical tractability and in terms of generalizability to the Multinomial case.