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Decompounding discrete distributions: A nonparametric Bayesian approach

Author

Listed:
  • Shota Gugushvili
  • Ester Mariucci
  • Frank van der Meulen

Abstract

Suppose that a compound Poisson process is observed discretely in time and assume that its jump distribution is supported on the set of natural numbers. In this paper we propose a nonparametric Bayesian approach to estimate the intensity of the underlying Poisson process and the distribution of the jumps. We provide a Markov chain Monte Carlo scheme for obtaining samples from the posterior. We apply our method on both simulated and real data examples, and compare its performance with the frequentist plug‐in estimator proposed by Buchmann and Grübel. On a theoretical side, we study the posterior from the frequentist point of view and prove that as the sample size n→∞, it contracts around the “true,” data‐generating parameters at rate 1/n, up to a logn factor.

Suggested Citation

  • Shota Gugushvili & Ester Mariucci & Frank van der Meulen, 2020. "Decompounding discrete distributions: A nonparametric Bayesian approach," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(2), pages 464-492, June.
    • Handle: RePEc:bla:scjsta:v:47:y:2020:i:2:p:464-492
    DOI: 10.1111/sjos.12413
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    References listed on IDEAS

    as
    1. Kappus, Johanna, 2014. "Adaptive nonparametric estimation for Lévy processes observed at low frequency," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 730-758.
    2. Shota Gugushvili & Frank Meulen & Peter Spreij, 2018. "A non-parametric Bayesian approach to decompounding from high frequency data," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 53-79, April.
    3. Zhang, Huiming & Liu, Yunxiao & Li, Bo, 2014. "Notes on discrete compound Poisson model with applications to risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 325-336.
    4. Ghosal,Subhashis & van der Vaart,Aad, 2017. "Fundamentals of Nonparametric Bayesian Inference," Cambridge Books, Cambridge University Press, number 9780521878265, January.
    5. Nickl, Richard & Reiß, Markus, 2012. "A Donsker theorem for Lévy measures," SFB 649 Discussion Papers 2012-003, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    6. Ole E. Barndorff-Nielsen & David G. Pollard & Neil Shephard, 2012. "Integer-valued L�vy processes and low latency financial econometrics," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 587-605, January.
    7. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
    8. Boris Buchmann & Rudolf Grübel, 2004. "Decompounding poisson random sums: Recursively truncated estimates in the discrete case," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 743-756, December.
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