Some properties of a generalized type-1 Dirichlet distribution

Abstract

This paper deals with a generalization of type-1 Dirichlet density by incorporating partial sums of the component variables. We study various proportions, structural decompositions, connections to random volumes and p -parallelotopes. We will also look into the regression function of x k on x 1 ,..., x k- 1 , Bayes’ estimates for the probabilities of a multinomial distribution by using this generalized Dirichlet model as the prior density are given. Other results illustrate the importance of the study of variable x 1 in this model. It is found that the variable x 1 in this model can be represented as the ratio of squares of volumes of two parallelotopes. Under certain conditions, x 1 can be used to study the structural representations of the likelihood ratio criteria in MANOVA, MANCOVA etc.