A family of consistent normally distributed tests for Poissonity

TL;DR

This paper introduces a new family of consistent, easy-to-implement tests for Poissonity based on probability generating functions, demonstrating good finite sample performance through extensive simulations.

Contribution

It proposes a novel family of tests for Poisson distribution that are simple, consistent, and based on the probability generating function characterization.

Findings

Test statistics are asymptotically normal.

The tests perform well compared to existing methods.

Finite sample properties are validated through simulations.

Abstract

A family of consistent tests, derived from a characterization of the probability generating function, is proposed for assessing Poissonity against a wide class of count distributions, which includes some of the most frequently adopted alternatives to the Poisson distribution. Actually, the family of test statistics is based on the difference between the plug-in estimator of the Poisson cumulative distribution function and the empirical cumulative distribution function. The test statistics have an intuitive and simple form and are asymptotically normally distributed, allowing a straightforward implementation of the test. The finite sample properties of the test are investigated by means of an extensive simulation study. The test shows satisfactory behaviour compared to other tests with known limit distribution.