On the Hyper-Poisson Distribution and its Generalization with Applications

Abstract

In this paper, we t the hyper-Poisson, and the Mittag-Leer function (MLFD) distributions to data exhibiting over and under dispersion. Three frequency data sets were employed with one exhibiting under-dispersion. We also extend these distributions to GLM situations where we have a set of covariates de ned in the form x′β. In all, we compared the negative-binomial (NB), the generalized Poisson (GP), the Conway-Maxwell Poisson (COMP), the Hyper-Poisson (HP) and the MLFD models to the selected data sets. The generalized linear model (GLM) data employed in this study is the German national health registry data which has 3874 observations with 41.56% being zeros-thus the data is zero-inflated. Our results contrast the results from these various distributions. Further, theoretical means and variances of each model are computed together with their corresponding empirical means and variances. It was obvious that the two do not match for each of our data sets. The reason being that the models all have infinite range of values than the random variable Y can take, but the data has a finite range of values. It is therefore not unusual for the sum of estimated probabilities being less than 1.00 and consequently, the sum of the expected values are usually less that the sample size n. However, if the range of values of Y are extended beyond the given data value,both theoretical and empirical moments as expected would be equal. We explore an alternative model for one of the data set. In contrast, most results in the literature sometimes just assume that the last category k has probability  which does not truly reflect the underlying probability structure from the data. We have employed SAS PROC NLMIXED in all our computations in this paper with the choice optimization algorithm being the conjugate gradient algorithm. We also computed the Wald test statistic for each data based on both the theoretical and empirical means and variances. Our results extend previous results on the analyzes of the chosen data in this example. Further, results obtained here indicate that some results in earlier studies on the data employed in this study may be in accurate. In others, our results are consistent with previous analyses on the data sets chosen for this article. While we do not pretend that the results obtained are entirely new, however, the analyses give opportunities to researchers in the eld the opportunity of implementing these models in SAS.