Studying the Third Cumulant of the Mixture of Dirichlet-multinomial Distributions

Abstract

Traditionally, the overdispersion parameter is estimated by using Pearson’s lack of fit statistic or the Deviance statistic , which do not perform well in the case of sparse data. This paper particularly focuses on an estimator of overdispersion parameter which was proposed for sparse multinomial data. The estimator was derived on the basis of an assumption on the 3rd cumulant of the response variable.When the data comes from the Dirichlet-multinomial distribution is known to have the lowest root mean squared error comparing to the other three estimators. In this paper the 1st to 3rd order raw moments of the finite mixture of Dirichlet-multinomial distributions are derived, which results in complicated mathematical expressions. Furthermore, it is found that the 3rd cumulant of this mixture does not satisfy the assumption which is considered in the derivation of .