Published

2013-07-01

Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors

Inferencia Bayesiana para la distribución Gamma de dos parámetros asumiendo diferentes a prioris no informativas

Keywords:

Gamma distribution, noninformative prior, copula, conjugate, Jeffreys prior, reference, MDIP, orthogonal, MCMC. (en)
a prioris de Jeffrey, a prioris no informativas, conjugada, cópulas, distribución Gamma, MCMC, MDIP, ortogonal, referencia (es)

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Authors

  • Fernando Antonio Moala Universidade Estadual Paulista
  • Pedro Luiz Ramos Universidade Estadual Paulista
  • Jorge Alberto Achcar Universidade de São Paulo

In this paper distinct prior distributions are derived in a Bayesian inference of the two-parameters Gamma distribution. Noniformative priors, such as Jeffreys, reference, MDIP, Tibshirani and an innovative prior based on the copula approach are investigated. We show that the maximal data information prior provides in an improper posterior density and that the different choices of the parameter of interest lead to different reference priors in this case. Based on the simulated data sets, the Bayesian estimates and credible intervals for the unknown parameters are computed and the performance of the prior distributions are evaluated. The Bayesian analysis is conducted using the Markov Chain Monte Carlo (MCMC) methods to generate samples from the posterior distributions under the above priors

En este artículo diferentes distribuciones a priori son derivadas en una inferencia Bayesiana de la distribución Gamma de dos parámetros. A prioris no informativas tales como las de Jeffrey, de referencia, MDIP, Tibshirani y una priori innovativa basada en la alternativa por cópulas son investigadas. Se muestra que una a priori de información de datos maximales conlleva a una a posteriori impropia y que las diferentes escogencias del parámetro de interés permiten diferentes a prioris de referencia en este caso. Datos simulados permiten calcular las estimaciones Bayesianas e intervalos de credibilidad para los parámetros desconocidos así como la evaluación del desempeño de las distribuciones a priori evaluadas. El análisis Bayesiano se desarrolla usando métodos MCMC (Markov Chain Monte Carlo) para generar las muestras de la distribución a posteriori bajo las a priori consideradas.

Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors

Inferencia Bayesiana para la distribución Gamma de dos parámetros asumiendo diferentes a prioris no informativas

FERNANDO ANTONIO MOALA1, PEDRO LUIZ RAMOS2, JORGE ALBERTO ACHCAR3

1Universidade Estadual Paulista, Facultad de Ciencia y Tecnología, Departamento de Estadística, Presidente Prudente, Brasil. Professor. Email: femoala@fct.unesp.br
2Universidade Estadual Paulista, Facultad de Ciencia y Tecnología, Departamento de Estadística, Presidente Prudente, Brasil. Student. Email: pedrolramos@hotmail.com
3Universidade de São Paulo, Facultad de Medicina de Ribeirão Preto, Departamento de Medicina Social, Ribeirão Preto, Brasil. Professor. Email: achcar@fmrp.usp.br

Abstract

In this paper distinct prior distributions are derived in a Bayesian inference of the two-parameters Gamma distribution. Noniformative priors, such as Jeffreys, reference, MDIP, Tibshirani and an innovative prior based on the copula approach are investigated. We show that the maximal data information prior provides in an improper posterior density and that the different choices of the parameter of interest lead to different reference priors in this case. Based on the simulated data sets, the Bayesian estimates and credible intervals for the unknown parameters are computed and the performance of the prior distributions are evaluated. The Bayesian analysis is conducted using the Markov Chain Monte Carlo (MCMC) methods to generate samples from the posterior distributions under the above priors.

Key words: Gamma distribution, noninformative prior, copula, conjugate, Jeffreys prior, reference, MDIP, orthogonal, MCMC.

Resumen

En este artículo diferentes distribuciones a priori son derivadas en una inferencia Bayesiana de la distribución Gamma de dos parámetros. A prioris no informativas tales como las de Jeffrey, de referencia, MDIP, Tibshirani y una priori innovativa basada en la alternativa por cópulas son investigadas. Se muestra que una a priori de información de datos maximales conlleva a una a posteriori impropia y que las diferentes escogencias del parámetro de interés permiten diferentes a prioris de referencia en este caso. Datos simulados permiten calcular las estimaciones Bayesianas e intervalos de credibilidad para los parámetros desconocidos así como la evaluación del desempeño de las distribuciones a priori evaluadas. El análisis Bayesiano se desarrolla usando métodos MCMC (Markov Chain Monte Carlo) para generar las muestras de la distribución a posteriori bajo las a priori consideradas.

Palabras clave: a prioris de Jeffrey, a prioris no informativas, conjugada, cópulas, distribución Gamma, MCMC, MDIP, ortogonal, referencia.

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References

1. Apolloni, B. & Bassis, S. (2099), 'Algorithmic inference of two-parameter gamma distribution', Communications in Statistics - Simulation and Computation 38(9), 1950-1968.

2. Berger, J. & Bernardo, J. M. (1992), On the development of the reference prior method, '', Fourth Valencia International Meeting on Bayesian Statistics, Spain.

3. Bernardo, J. M. (1979), 'Reference posterior distributions for Bayesian inference', Journal of the Royal Statistical Society 41(2), 113-147.

4. Cox, D. R. & Reid, N. (1987), 'Parameter orthogonality and approximate conditional inference (with discussion)', Journal of the Royal Statistical Society, Series B 49, 1-39.

5. Gelfand, A. E. & Smith, F. M. (1990), 'Sampling-based approaches to calculating marginal densities', Journal of the American Statistical Association 85, 398-409.

6. Gilks, W., Clayton, D., Spiegelhalter, D., Best, N., McNiel, A., Sharples, L. & Kirby, A. (1993), 'Modeling complexity: applications of Gibbs sampling in medicine', Journal of the Royal Statistical Society, Series B 55(1), 39-52.

7. Jeffreys, S. H. (1967), Theory of Probability, 3 edn, Oxford University Press, London.

8. Lawless, J. (1982), Statistical Models and Methods for Lifetime Data, John Wiley, New York.

9. Min, C-k & Zellner, A. (1993), Bayesian Analysis, Model Selection and Prediction, 'Physics and Probability: Essays in honor of Edwin T Jaynes', Cambridge University Press, p. 195-206.

10. Moala, F. (2010), 'Bayesian analysis for the Weibull parameters by using noninformative prior distributions', Advances and Applications in Statistics(14), 117-143.

11. Morgenstern, D. (1956), 'Einfache beispiele sw edimensionaler vertielung', Mit Mathematics Statistics 8, 234-235.

12. Nelsen, R. B. (1999), An Introduction to Copulas, Springer Verlag, New York.

13. Pradhan, B. & Kundu, D. (2011), 'Bayes estimation and prediction of the two-parameter Gamma distribution', Journal of Statistical Computation and Simulation 81(9), 1187-1198.

14. Smith, A. & Roberts, G. (1993), 'Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo Methods', Journal of the Royal Statistical Society: Series B 55, 3-24.

15. Son, Y. & Oh, M. (2006), 'Bayesian estimation of the two-parameter Gamma distribution', Communications in Statistics - Simulation and Computation 35, 285-293.

16. Tibshirani, R. (1989), 'Noninformative prioris for one parameters of many', Biometrika 76, 604-608.

17. Trivedi, P. K. & Zimmer, D. M. (2005a), 'Copula modelling: an introduction to practicioners', Foundations and Trends in Econometrics(1), 1-111.

18. Trivedi, P. K. & Zimmer, D. M. (2005b), Copula Modelling, New Publishers, New York.

19. Zellner, A. (1977), Maximal data information prior distributions, 'In New Methods in the Applications of Bayesian Methods', North-Holland, Amsterdam.

20. Zellner, A. (1984), Maximal Data Information Prior Distributions, Basic Issues in Econometrics, The University of Chicago Press, Chicago, USA.

21. Zellner, A. (1990), Bayesian methods and entropy in economics and econometrics, 'Maximum Entropy and Bayesian Methods', Dordrecht, Netherlands: Kluwer Academic Publishers, p. 17-31.

[Recibido en enero de 2013. Aceptado en septiembre de 2013]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv36n2a09,
    AUTHOR  = {Antonio Moala, Fernando and Luiz Ramos, Pedro and Alberto Achcar, Jorge},
    TITLE   = {{Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2013},
    volume  = {36},
    number  = {2},
    pages   = {319-336}
}

References

Apolloni, B. & Bassis, S. (2099), ‘Algorithmic inference of two-parameter gamma distribution’, Communications in Statistics - Simulation and Computation 38(9), 1950–1968.

Berger, J. & Bernardo, J. M. (1992), On the development of the reference prior method, Fourth Valencia International Meeting on Bayesian Statistics, Spain.

Bernardo, J. M. (1979), ‘Reference posterior distributions for Bayesian inference’, Journal of the Royal Statistical Society 41(2), 113–147.

Cox, D. R. & Reid, N. (1987), ‘Parameter orthogonality and approximate conditional inference (with discussion)’, Journal of the Royal Statistical Society, Series B 49, 1–39.

Gelfand, A. E. & Smith, F. M. (1990), ‘Sampling-based approaches to calculating marginal densities’, Journal of the American Statistical Association 85, 398–409.

Gilks, W., Clayton, D., Spiegelhalter, D., Best, N., McNiel, A., Sharples, L. & Kirby, A. (1993), ‘Modeling complexity: Applications of Gibbs sampling in medicine’, Journal of the Royal Statistical Society, Series B 55(1), 39–52.

Jeffreys, S. H. (1967), Theory of Probability, 3 edn, Oxford University Press, London.

Lawless, J. (1982), Statistical Models and Methods for Lifetime Data, John Wiley, New York.

Min, C.-k. & Zellner, A. (1993), Bayesian Analysis, Model Selection and Prediction, in ‘Physics and Probability: Essays in honor of Edwin T Jaynes’, Cambridge University Press, pp. 195–206.

Moala, F. (2010), ‘Bayesian analysis for the Weibull parameters by using noninformative prior distributions’, Advances and Applications in Statistics (14), 117–143.

Morgenstern, D. (1956), ‘Einfache beispiele sw edimensionaler vertielung’, Mit Mathematics Statistics 8, 234–235.

Nelsen, R. B. (1999), An Introduction to Copulas, Springer Verlag, New York.

Pradhan, B. & Kundu, D. (2011), ‘Bayes estimation and prediction of the twoparameter Gamma distribution’, Journal of Statistical Computation and Simulation 81(9), 1187–1198.

Smith, A. & Roberts, G. (1993), ‘Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo Methods’, Journal of the Royal Statistical Society: Series B 55, 3–24.

Son, Y. & Oh, M. (2006), ‘Bayesian estimation of the two-parameter Gamma distribution’, Communications in Statistics – Simulation and Computation 35, 285–293.

Tibshirani, R. (1989), ‘Noninformative prioris for one parameters of many’, Biometrika 76, 604–608.

Trivedi, P. K. & Zimmer, D. M. (2005a), Copula Modelling, New Publishers, New York.

Trivedi, P. K. & Zimmer, D. M. (2005b), ‘Copula modelling: An introduction to practicioners’, Foundations and Trends in Econometrics (1), 1–111.

Zellner, A. (1977), Maximal data information prior distributions, in A. Aykac & C. Brumat, eds, ‘In New Methods in the Applications of Bayesian Methods’, North-Holland, Amsterdam.

Zellner, A. (1984), Maximal Data Information Prior Distributions, Basic Issues in Econometrics, The University of Chicago Press, Chicago, USA.

Zellner, A. (1990), Bayesian methods and entropy in economics and econometrics, in W. J. Grandy & L. Schick, eds, ‘Maximum Entropy and Bayesian Methods’, Dordrecht, Netherlands: Kluwer Academic Publishers, pp. 17–31.

How to Cite

APA

Moala, F. A., Ramos, P. L. & Achcar, J. A. (2013). Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors. Revista Colombiana de Estadística, 36(2), 319–336. https://revistas.unal.edu.co/index.php/estad/article/view/44351

ACM

[1]
Moala, F.A., Ramos, P.L. and Achcar, J.A. 2013. Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors. Revista Colombiana de Estadística. 36, 2 (Jul. 2013), 319–336.

ACS

(1)
Moala, F. A.; Ramos, P. L.; Achcar, J. A. Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors. Rev. colomb. estad. 2013, 36, 319-336.

ABNT

MOALA, F. A.; RAMOS, P. L.; ACHCAR, J. A. Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors. Revista Colombiana de Estadística, [S. l.], v. 36, n. 2, p. 319–336, 2013. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/44351. Acesso em: 16 nov. 2025.

Chicago

Moala, Fernando Antonio, Pedro Luiz Ramos, and Jorge Alberto Achcar. 2013. “Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors”. Revista Colombiana De Estadística 36 (2):319-36. https://revistas.unal.edu.co/index.php/estad/article/view/44351.

Harvard

Moala, F. A., Ramos, P. L. and Achcar, J. A. (2013) “Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors”, Revista Colombiana de Estadística, 36(2), pp. 319–336. Available at: https://revistas.unal.edu.co/index.php/estad/article/view/44351 (Accessed: 16 November 2025).

IEEE

[1]
F. A. Moala, P. L. Ramos, and J. A. Achcar, “Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors”, Rev. colomb. estad., vol. 36, no. 2, pp. 319–336, Jul. 2013.

MLA

Moala, F. A., P. L. Ramos, and J. A. Achcar. “Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors”. Revista Colombiana de Estadística, vol. 36, no. 2, July 2013, pp. 319-36, https://revistas.unal.edu.co/index.php/estad/article/view/44351.

Turabian

Moala, Fernando Antonio, Pedro Luiz Ramos, and Jorge Alberto Achcar. “Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors”. Revista Colombiana de Estadística 36, no. 2 (July 1, 2013): 319–336. Accessed November 16, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/44351.

Vancouver

1.
Moala FA, Ramos PL, Achcar JA. Bayesian Inference for Two-Parameter Gamma Distribution Assuming Different Noninformative Priors. Rev. colomb. estad. [Internet]. 2013 Jul. 1 [cited 2025 Nov. 16];36(2):319-36. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/44351

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