Choice between and within the classes of Poisson-Tweedie and Poisson-exponential-Tweedie count models

In both flexible Poisson Tweedie (PT) and Poisson-exponential-Tweedie (PET) over-dispersed count models, the common power parameter p is an element of {0} boolean OR [1, infinity) works as an automatic distribution selection. It mainly separates two subclasses of zero-inflated (1 <= p < 2) and heavy-tailed (p > 2). Although extensive works have been conducted in discriminating between continuous and semicontinuous distribution functions, not much attention has been paid to discriminating between discrete distribution functions. Estimations based on the likelihood approach for PT and PETS are challenging owing to the presence of an infinity sum and intractable integrals in their probability mass functions. Thus, we attempt to perform a Monte Carlo simulation to approximate them. In this paper, we invest the ratio of the maximized likelihoods within each subclass of these two models for discrimination purposes based on classic measures of dispersion, zero-inflation and heavy tail of both PT and PET models. Grounded only on dispersion index, we also discriminate two particular cases of the Hermite distribution with respect to its geometric version (p = 0) and the negative binomial one with its geometric version (p = 2). Probabilities of correct selection for several combinations of dispersion parameters, means and sample sizes are examined by simulations. We carry out a numerical comparison study to assess the discrimination procedures in these subclasses of two families.