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Exact Simulation of Poisson-Dirichlet Distribution and Generalised Gamma Process

Author

Listed:
  • Angelos Dassios

    (London School of Economics)

  • Junyi Zhang

    (The Hong Kong Polytechnic University)

Abstract

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.

Suggested Citation

  • Angelos Dassios & Junyi Zhang, 2023. "Exact Simulation of Poisson-Dirichlet Distribution and Generalised Gamma Process," Methodology and Computing in Applied Probability, Springer, vol. 25(2), pages 1-21, June.
  • Handle: RePEc:spr:metcap:v:25:y:2023:i:2:d:10.1007_s11009-023-10040-3
    DOI: 10.1007/s11009-023-10040-3
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    References listed on IDEAS

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    1. Ilenia Epifani, 2003. "Exponential functionals and means of neutral-to-the-right priors," Biometrika, Biometrika Trust, vol. 90(4), pages 791-808, December.
    2. François Caron & Emily B. Fox, 2017. "Sparse graphs using exchangeable random measures," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(5), pages 1295-1366, November.
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