Some remarks on testing goodness of fit for the poisson assumption

Abstract

The primary objective of this note is to develop a locally optimal test for detecting negative binomial departure from the Poisson distribution. Specifically, we consider as an alternative of the Poisson a negative binomial distribution for which the variance is a linear function of the mean. This research is motivated by Collings and Margolin’s work (1985) on the detection of negative binomial departure from the Poisson assumption when the variance is a quadratic function of the mean. We develop locally optimal tests in the regression-through-the-origin and the one-way layout cases. We further show that in our set up and also in the context of a negative binomial distribution with variance neither linear nor quadratic in the mean tests based on our statistics, say SB and SC are asymptotically superior to tests based on Collings and Margolin’s, say TB and TC respectively. This asymptotic superiority is supported by Monte Carlo simulation studies. Our results show that the optimal tests for detecting negative binomial departure from the Poisson assumption depend on the mean and variance structure of the negative binomial distribution.