A New Versatile Discrete Distribution

Abstract

This paper introduces a new flexible distribution for discrete data. Approximate moment estimators of the parameters of the distribution, to be used as starting values for numerical optimization procedures, are discussed. “Exact” moment estimation, effected via a numerical procedure, and maximum likelihood estimation, are considered. The quality of the results produced by these estimators is assessed via simulation experiments. Several examples are given of fitting instances of the new distribution to real and simulated data. It is noted that the new distribution is a member of the exponential family. Expressions for the gradient and Hessian of the log-likelihood of the new distribution are derived. The former facilitates the numerical maximization of the likelihood with optim(); the latter provides means of calculating or estimating the covariance matrix of of the parameter estimates. A discrepancy between estimates of the covariance matrix obtained by inverting the Hessian and those obtained by Monte Carlo methods is discussed.