From Beta to Dirichlet Frontiers

Abstract

In this study, two classes of multivariate distributions are proposed as extensions of the well known univariate class of beta-generated distributions. This extension from the univariate to the multivari- ate domain addresses the need of flexible multivariate distributions that can model a wide range of multivariate data sets where outliers are present. The first class is constructed by embedding the cumulative distribution functions (cdf) of univariate baseline distributions within the probability den- sity function (pdf) of the Dirichlet type I distribution. The second class is constructed through an interesting view of embedding the cdf of a multivariate distribution within the pdf of the univariate beta distribution. Each class presents their own unique properties such as specific parameter require- ments and dependence structures for distributions belonging to these classes. An example of a newly developed distribution for each class is investigated, where the value and performance of the models are illustrated using real data sets and simulation studies. The method of maximum likelihood is used for parameter estimation; and measures such as the Kolmogorov-Smirnov distance is implemented as a performance based measure for competing models. A new model testing technique is also introduced to evaluate the performance of the multivariate models. Possible extensions of these classes of distributions are discussed for future research.