Goodness-of-fit tests for the bivariate Poisson distribution

Abstract

The bivariate Poisson distribution is commonly used to model bivariate count data. In this paper we study a goodness-of-fit test for this distribution. We also provide a review of the existing tests for the bivariate Poisson distribution, and its multivariate extension. The proposed test is consistent against any fixed alternative. It is also able to detect local alternatives converging to the null at the rate $n^{-\frac{1}{2}}.$ The bootstrap can be employed to consistently estimate the null distribution of the test statistic. Through a simulation study we investigated the goodness of the bootstrap approximation and the power for finite sample sizes.

Keywords:

• Bivariate Poisson distribution
• Goodness-of-fit
• Empirical probability generating function
• Consistency against fixed alternatives
• Bootstrap distribution estimator

Acknowledgments

The author would like to thank the Departamento de Investigación de la Universidad del Bío-Bío and the Grupo de Investigación Matemática Aplicada GI 172409/C de la Universidad del Bío-Bío, Chile. He also thanks the anonymous reviewers and the editor of this journal for their valuable time and their careful comments and suggestions with which the quality of this paper has been improved. Thanks to Florencia Osorio for her valuable comments in the revision of the manuscript.