A Two-Parameter Family of Hyper-Poisson Distributions

A two-parameter generalization of the Poisson distributions, called the “hyper-Poisson” family, is derived and discussed. It is in turn a special case of the three-parameter confluent hypergeometric series distributions. When one parameter is fixed, the hyper-Poisson distributions are also power series distributions. The distributions are classified as “sub-Poisson” or “super-Poisson” according as the variance is less than or greater than the mean. The latter category includes the truncated Poisson distributions. Certain properties of the hyper-Poisson distributions are considered, including moment, modified moment, and maximum likelihood estimators of the parameters. Estimation using moments is illustrated on Student's haemacytometer data.