A Novel Approach to Handling the Non-Central Dirichlet Distribution

TL;DR

This paper introduces a new approach to analyze the Non-central Dirichlet distribution, simplifying its complex joint density and deriving a new stochastic representation and closed-form moments.

Contribution

A novel method based on conditional density and property transposition that simplifies the analysis of the Non-central Dirichlet distribution.

Findings

Derived a simple stochastic representation of the distribution.

Obtained a closed-form expression for mixed raw moments.

Provided insights that simplify the mathematical complexity.

Abstract

In the present paper new insights into the study of the Non-central Dirichlet distribution are provided. This latter is the analogue of the Dirichlet distribution obtained by replacing the Chi-Squared random variables involved in its definition by as many non-central ones. Specifically, a novel approach to tackling the analysis of this model is introduced based on a simple conditional density together with a suitable transposition into the non-central framework of a characterizing property of independent Chi-Squared random variables. This approach thus enables to remedy the undeniable mathematical complexity of the joint density function of such distribution by paving the way towards achieving a new attractive stochastic representation as well as a surprisingly simple closed-form expression for its mixed raw moments.