  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.
