A fractional generalization of the Dirichlet distribution and related distributions

TL;DR

This paper introduces a fractional generalization of the Dirichlet distribution based on Mittag-Leffler distributed partitions, deriving its properties and exploring related distributions with simulation results.

Contribution

It presents a novel fractional Dirichlet distribution derived from Mittag-Leffler assumptions and analyzes its properties and relations to existing generalizations.

Findings

Derived the expected value and variance of the marginal distribution.

Provided the probability density function for the fractional Dirichlet.

Conducted Monte Carlo simulations to illustrate the distribution's behavior.

Abstract

This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the $n$ partitions of the interval $[0,W_n]$ are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.