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

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 \(O(a^{n})\), \(a>1\), 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 \(r=1,\ldots ,L\) 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.

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Acknowledgements

E. Moreno and F.J. Vázquez–Polo was partially funded by Grant ECO2017–85577–P from the Spanish Ministry of Economy and Competitiveness. The authors thank the Associate Editor and two referees for their suggestions which have improved the original manuscript.

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Correspondence to Francisco–José Vázquez–Polo.

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EM and FJVP partially funded by Grant ECO2017–85577–P from the Spanish Ministry of Economy and Competitiveness (MINECO, Spain). The authors thank the Associate Editor and two referees for their suggestions which have improved the original manuscript.

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Moreno, E., Martínez, C. & Vázquez–Polo, F. Objective Bayesian model choice for non-nested families: the case of the Poisson and the negative binomial. TEST 30, 255–273 (2021). https://doi.org/10.1007/s11749-020-00717-z

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Keywords

  • Bayesian model selection
  • Test for separate families
  • Consistency
  • Sampling behavior for small sample sizes
  • Rate of convergence

Mathematics Subject Classification

  • 62F15
  • 62F03
  • 62P05