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Objective Bayesian model choice for non-nested families: the case of the Poisson and the negative binomial

Author

Listed:
  • Elías Moreno

    (University of Granada)

  • Carmen Martínez

    (University of Granada)

  • Francisco–José Vázquez–Polo

    (University of Las Palmas de Gran Canaria)

Abstract

Selecting a statistical model from a set of competing models is a central issue in the scientific task, and the Bayesian approach to model selection is based on the posterior model distribution, a quantification of the updated uncertainty on the entertained models. We present a Bayesian procedure for choosing a family between the Poisson and the geometric families and prove that the procedure is consistent with rate $$O(a^{n})$$ O ( a n ) , $$a>1$$ a > 1 , where a is a function of the parameter of the true model. An extension of this procedure to the multiple testing Poisson and negative binomial with r successes for $$r=1,\ldots ,L$$ r = 1 , … , L is also proved to be consistent with exponential rate. For small sample sizes, a simulation study indicates that the model selection between the above distributions is made with large uncertainty when sampling from a specific subset of distributions. This difficulty is however mitigated by the large consistency rate of the procedure.

Suggested Citation

  • Elías Moreno & Carmen Martínez & Francisco–José Vázquez–Polo, 2021. "Objective Bayesian model choice for non-nested families: the case of the Poisson and the negative binomial," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(1), pages 255-273, March.
  • Handle: RePEc:spr:testjl:v:30:y:2021:i:1:d:10.1007_s11749-020-00717-z
    DOI: 10.1007/s11749-020-00717-z
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    References listed on IDEAS

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    1. Dawid, A. Philip & Musio, Monica & Columbu, Silvia, 2017. "A note on Bayesian model selection for discrete data using proper scoring rules," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 101-106.
    2. Jean-Yves Dauxois & Pierre Druilhet & Denys Pommeret, 2006. "A bayesian choice between poisson, binomial and negative binomial models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(2), pages 423-432, September.
    3. D. R. Cox, 2013. "A return to an old paper: ‘Tests of separate families of hypotheses’," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(2), pages 207-215, March.
    4. Liang, Feng & Paulo, Rui & Molina, German & Clyde, Merlise A. & Berger, Jim O., 2008. "Mixtures of g Priors for Bayesian Variable Selection," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 410-423, March.
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