Exact Simulation of Poisson-Dirichlet Distribution and Generalised Gamma Process

Let $$J_1>J_2>\dots $$ J 1 > J 2 > ⋯ be the ranked jumps of a gamma process $$\tau _{\alpha }$$ τ α on the time interval $$[0,\alpha ]$$ [ 0 , α ] , such that $$\tau _{\alpha }=\sum _{k=1}^{\infty }J_k$$ τ α = ∑ k = 1 ∞ J k . In this paper, we design an algorithm that samples from the random vector $$(J_1, \dots , J_N, \sum _{k=N+1}^{\infty }J_k)$$ ( J 1 , ⋯ , J N , ∑ k = N + 1 ∞ 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.