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

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.

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.         [ Links ]

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

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

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.         [ Links ]

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

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.         [ Links ]

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

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

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.         [ Links ]

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

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

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

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.         [ Links ]

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.         [ Links ]

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

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

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

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

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

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

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.         [ Links ]

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