Exact Simulation of Poisson-Dirichlet Distribution and Generalised Gamma Process

Abstract

Let J_1>J_2>dots be the ranked jumps of a gamma process tau _{alpha } on the time interval [0,alpha ], such that tau _{alpha }=sum _{k=1}^{infty }J_k. In this paper, we design an algorithm that samples from the random vector (J_1, dots , J_N, sum _{k=N+1}^{infty }J_k). Our algorithm provides an analog to the well-established inverse Lévy measure (ILM) algorithm by replacing the numerical inversion of exponential integral with an acceptance-rejection step. This research is motivated by the construction of Dirichlet process prior in Bayesian nonparametric statistics. The prior assigns weight to each atom according to a GEM distribution, and the simulation algorithm enables us to sample from the N largest random weights of the prior. Then we extend the simulation algorithm to a generalised gamma process. The simulation problem of inhomogeneous processes will also be considered. Numerical implementations are provided to illustrate the effectiveness of our algorithms.