Generalized Rank Dirichlet distributions

We study a new parametric family of distributions on the ordered simplex del(d-1) = {y is an element of R-d) : y(1) >= ... >=: y(d) >= 0, Sigma(d)(k=1) y(k) = 1}, which we call Generalized Rank Dirichlet (GRD) distributions. Their density is proportional to Pi(d)(k=1) y(k)(ak-1) for a parameter a = (a(1),.,...a(d)) is an element of R-d satisfying a(k) + a(k+1) + ... + a(d) > 0 for k = 2, .,..d. The density is similar to the Dirichlet distribution, but is defined on del(d-1), leading to different properties. In particular, certain components a(k) can be negative. Random variables Y = (Y-1,.,..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.. explicit expressions for moments of order M is an element of N for the Y-k's and moments of all orders for the log gaps Z(k) = log Yk-1 - log Y-k when a(1) + ... + a(d) = -M. Additionally, we propose an algorithm to exactly simulate random variates in this case. In the general case a(1) + ... + a(d) is an element of R we obtain series representations for these quantities and provide an approximate simulation algorithm.

Automated summary

In this study, we examine a new family of distributions called Generalized Rank Dirichlet distributions on the ordered simplex. We investigate their properties and propose simulation algorithms for random variates. The results can be applied to model capital distribution in financial markets and ranked order statistics of weight vectors.