Goodness-of-Fit Tests for Additively Closed Count Models with an Application to the Generalized Hermite Distribution

摘要

A general method is proposed for testing the fit to any member of the invariant under convolutions family of count models, parameterized by mean and variance. The test statistics, which are of the weighted L2-type, exploit the specific structure of the corresponding probability generating function. Their asymptotic null distribution is obtained, and the consistency of the tests is studied. As an example, the generalized Hermite distribution, a specific member of this family, is analysed. In this case, two methods of estimation of the parameters are considered, for which limit statistics are obtained as the decay of the weight function tends to infinity. The performance of a parametric bootstrap version of the method is investigated via Monte Carlo.