A power series distribution involving the double factorial

Abstract

This work proposes a one–parameter distribution with a formulation that involves the double factorial. The new distribution is intended to be competitive with the Poisson, geometric and negative binomial distributions and, like these, belongs to the natural exponential family of distributions (EFD) and to power series distributions. The new distribution is over–dispersed and appears to be unimodal or multimodal depending of the value of its parameter. The behaviour of the rate of successive probabilities is different from that of traditional discrete distributions, in which this rate is always increasing or decreasing. Furthermore, the hazard rate function also presents unusual behaviour in comparison with the classical distribution types mentioned above. We establish the normal approximation to the distribution proposed here. Estimation methods are considered, and applications show that the distribution works well when the data sets considered present two or more modal values.