Exact simulation of Poisson-Dirichlet distribution and generalised gamma process

Abstract

Let J1 > J2 >... be the ranked jumps of a gamma process τα on the time interval [0,α], such that τα = ∞ k=1 Jk.Inthispaper, wedesignanalgorithm that samples from the random vector (J1,...,JN, ∞ k=N+1 Jk). 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.