# Generalized Rank Dirichlet Distributions

## TL;DR

This paper introduces the Generalized Rank Dirichlet (GRD) distributions, a new family of distributions on the ordered simplex that generalize the Dirichlet distribution, allowing for negative parameters and providing explicit moments and simulation methods.

## Contribution

The paper defines GRD distributions on the ordered simplex, derives explicit moments, and develops exact and approximate simulation algorithms, expanding modeling capabilities for ranked data.

## Key findings

- Explicit moments for GRD distributions across dimensions.
- Series representations and simulation algorithms for the distributions.
- Application potential in financial modeling and ranked statistics.

## Abstract

We study a new parametric family of distributions on the ordered simplex $\nabla^{d-1} = \{y \in \mathbb{R}^d: y_1 \geq \dots \geq y_d \geq 0, \sum_{k=1}^d y_k = 1\}$, which we call Generalized Rank Dirichlet (GRD) distributions. Their density is proportional to $\prod_{k=1}^d y_k^{a_k-1}$ for a parameter $a = (a_1,\dots,a_d) \in \mathbb{R}^d$ satisfying $a_k + a_{k+1} + \dots + a_d > 0$ for $k=2,\dots,d$. The density is similar to the Dirichlet distribution, but is defined on $\nabla^{d-1}$, leading to different properties. In particular, certain components $a_k$ can be negative. Random variables $Y = (Y_1,\dots,Y_d)$ with GRD distributions have previously been used to model capital distribution in financial markets and more generally can be used to model ranked order statistics of weight vectors. We obtain for any dimension $d$ explicit expressions for moments of order $M \in \mathbb{N}$ for the $Y_k$'s and moments of all orders for the log gaps $Z_k = \log Y_{k-1} - \log Y_k$ when $a_1 + \dots + a_d = -M$. Additionally, we propose an algorithm to exactly simulate random variates in this case. In the general case $a_1 + \dots + a_d \in \mathbb{R}$ we obtain series representations for these quantities and provide an approximate simulation algorithm.