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A general guide in Bayesian and robust Bayesian estimation using Dirichlet processes

Author

Listed:
  • Ali Karimnezhad

    (University of Ottawa)

  • Mahmoud Zarepour

    (University of Ottawa)

Abstract

In this paper, we investigate Bayesian and robust Bayesian estimation of a wide range of parameters of interest in the context of Bayesian nonparametrics under a broad class of loss functions. Dealing with uncertainty regarding the prior, we consider the Dirichlet and the Dirichlet invariant priors, and provide explicit form of the resulting Bayes and robust Bayes estimators. Tractability of the results is supported by numerous examples of different well-known loss functions. The practical utility of the proposed Bayes and robust Bayes estimators are examined for a real data set.

Suggested Citation

  • Ali Karimnezhad & Mahmoud Zarepour, 2020. "A general guide in Bayesian and robust Bayesian estimation using Dirichlet processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(3), pages 321-346, April.
    • Handle: RePEc:spr:metrik:v:83:y:2020:i:3:d:10.1007_s00184-019-00737-2
    DOI: 10.1007/s00184-019-00737-2
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    References listed on IDEAS

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    1. Ali Karimnezhad & Ahmad Parsian, 2014. "Robust Bayesian methodology with applications in credibility premium derivation and future claim size prediction," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(3), pages 287-303, July.
    2. James Berger & Elías Moreno & Luis Pericchi & M. Bayarri & José Bernardo & Juan Cano & Julián Horra & Jacinto Martín & David Ríos-Insúa & Bruno Betrò & A. Dasgupta & Paul Gustafson & Larry Wasserman &, 1994. "An overview of robust Bayesian analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 3(1), pages 5-124, June.
    3. Lijoi, Antonio & Mena, Ramses H. & Prunster, Igor, 2005. "Hierarchical Mixture Modeling With Normalized Inverse-Gaussian Priors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1278-1291, December.
    4. Lancelot F. James & Antonio Lijoi & Igor Prünster, 2006. "Conjugacy as a Distinctive Feature of the Dirichlet Process," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(1), pages 105-120, March.
    5. Fernando A. Quintana & Wesley O. Johnson & L. Elaine Waetjen & Ellen B. Gold, 2016. "Bayesian Nonparametric Longitudinal Data Analysis," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1168-1181, July.
    6. Fabrizio Ruggeri, 2014. "On Some Optimal Bayesian Nonparametric Rules for Estimating Distribution Functions," Econometric Reviews, Taylor & Francis Journals, vol. 33(1-4), pages 289-304, June.
    7. Ali Karimnezhad & Ahmad Parsian, 2019. "Bayesian and robust Bayesian analysis in a general setting," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(15), pages 3899-3920, August.
    8. Makov, Udi E., 1995. "Loss robustness via Fisher-weighted squared-error loss function," Insurance: Mathematics and Economics, Elsevier, vol. 16(1), pages 1-6, April.
    9. Dalal, S. R., 1979. "Dirichlet invariant processes and applications to nonparametric estimation of symmetric distribution functions," Stochastic Processes and their Applications, Elsevier, vol. 9(1), pages 99-107, August.
    10. Leila Golparver & Ali Karimnezhad & Ahmad Parsian, 2013. "Optimal rules and robust Bayes estimation of a Gamma scale parameter," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(5), pages 595-622, July.
    11. Kiapour, A. & Nematollahi, N., 2011. "Robust Bayesian prediction and estimation under a squared log error loss function," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1717-1724, November.
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