Generalized Rank Dirichlet Distributions

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.