On a new class of tests for the Poisson distribution based on empirical weight functions

Abstract

The aim of our study is to introduce a new class of tests for the Poisson distribution. The testing procedure entails fitting a weighted Poisson distribution to observed data. The Poisson distribution is a special case of this more general class of distributions. If the data are realised from a Poisson distribution, then the weight function associated with the weighted Poisson distribution is expected to be a constant (since a weighted Poisson distribution with a constant weight function is simply a Poisson distribution). The proposed test statistics are weighted Lp distances between an empirical version of the weight function and the corresponding constant. Computational forms are derived for test statistics based on weighted L1, L2 and L1 distances for three choices of weight function in each case. The tests considered reject the assumption that the observed data are Poisson for large values of these distance measures. A Monte Carlo study is included in which the finite sample performance of the newly proposed class of tests is investigated and compared to a wide range of existing tests for the Poisson distribution. The numerical powers are obtained using a parametric bootstrap approach known as the warp-speed bootstrap. The finite sample results indicate that the newly proposed tests are competitive in terms of their power performance. In fact, these tests often outperform the existing tests for the Poisson distribution considered, especially in the case of overdispersed alternatives (where the variance of the alternative distribution considered exceeds its mean). The study is concluded with the analysis of two observed datasets. Here we demonstrate the use of the newly proposed tests to test the hypothesis that a given dataset is realised from a Poisson distribution.