Published

2015-07-01

Bayesian Analysis of the 3-Component Mixture of Exponential Distribution Assuming the Non-Informative Priors

Análisis bayesiano de una mezcla de tres componentes de distribuciones exponenciales asumiéndolas a priori no informativas

DOI:

https://doi.org/10.15446/rce.v38n2.51670

Keywords:

Mixture Model, Bayes Estimators, Exponential Distribution, Loss Function, Posterior Risks (en)
Modelo de mezcla, Estimadores de Bayes, Distribución exponencial, Función de pérdida, Riesgos posteriores. (es)

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Authors

  • Muhammad Tahir Quaid-i-Azam University, Islamabad, Pakistan Government College University, Faisalabad, Pakistan
  • Muhammad Aslam Riphah International University, Islamabad, Pakistan
Bayesian analysis of the 3-component mixture of an Exponential distribution under type-I right censoring scheme is considered in this paper. The Bayes estimators and posterior risks for the unknown parameters are derived under squared error loss function, precautionary loss function and DeGroot loss function assuming the non-informative (uniform and Jeffreys') priors. The Bayes estimators and posterior risks are viewed as a function of the test termination time. A simulation study is given to highlight and compare the properties of the Bayes estimates.

El análisis bayesiano de una mezcla de tres componentes de una distribución exponencial bajo el esquema de censura a la derecha tipo I se considera en este artículo. Los estimadores de Bayes y los riesgos posteriores de los parámetros desconocidos son derivados bajo una función de pérdida de error cuadrático, función de perdida precautelary función de perdida de DeGroot asumiendo a prioris no informativas (uniforme y Jeffreys). Los estimadores de Bayes y los riesgos posteriores seven como una función del tiempo de terminación del test. Un estudio de simulación muestra y compara las propiedades de los estimadores de Bayes.

References

Abu-Taleb, A. A., Smadi, M. M. & Alawneh, A. J. (2007), ‘Bayes estimation of the lifetime parameters for the exponential distribution’, Journal of Mathematics and Statistics 3(3), 106–108.

Ali, M. M., Woo, J. & Nadarajah, S. (2005), ‘Bayes estimators of the exponential distribution’, Journal of Statistics and Management Systems 8(1), 53–58.

Bayes, T. (1763), ‘An essay toward solving a problem in the doctrine of chances’, Philosophical Transactions of the Royal Society of London 53, 370–418.

Berger, J. O. (1985), Statistical Decision Theory and Bayesian Analysis, 2 edn, Springer, New York.

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

Chin-Chuan, W., Liang-Yuh, O. & Hsin-Lin, K. (2003), ‘Bayesian estimations of some process capability indices under restrictive assumptions’, Journal of the Chinese Institute of Industrial Engineers 20(1), 49–61.

de Laplace, P. S. (1820), Théorie Analytique des Probabilités, Courcier, Paris.

DeGroot, M. H. (2005), Optimal Statistical Decisions, Vol. 82, John Wiley & Sons.

Geisser, S. (1984), ‘On prior distributions for binary trials’, The American Statistician 38(4), 244–247.

Gijbels, I. (2010), ‘Censored data’, Wiley Interdisciplinary Reviews: Comput Stat 2(2), 178–188.

Harris, C. M. (1983), ‘On finite mixtures of geometric and negative binomial distributions’, Communications in Statistics-Theory and Methods 12(9), 987–1007.

Hebert, J. L. & Scariano, S. M. (2005), ‘Comparing location estimators for exponential mixtures under pitman’s measure of closeness’, Communications in Statistics-Theory and Methods 33(1), 29–46.

Jeffreys, H. (1946), ‘An invariant form for the prior probability in estimation problems’, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 186(1007), 453–461.

Jeffreys, H. (1961), Theory of Probability, Claredon Press, Oxford, UK.

Jones, P. N. & McLachlan, G. J. (1990), ‘Laplace-normal mixtures fitted to wind shear data’, Journal of Applied Statistics 17(2), 271–276.

Kalbfleisch, J. D. & Prentice, R. L. (2011), The statistical Analysis of Failure Time Data, Vol. 360, John Wiley & Sons.

Kanji, G. K. (1985), ‘A mixture model for wind shear data’, Journal of Applied Statistics 12(1), 49–58.

Kazmi, S. M. A., Aslam, M. & Ali, S. (2012), ‘On the Bayesian estimation for two component mixture of Maxwell distribution, assuming type I censored data’, International Journal of Applied Science and Technology 2(1), 197–218.

Legendre, A. M. (1806), Nouvelles méthodes pour la determination des orbites des cometes. (Apéndice: Sur la méthode des moindres carrés), Courcier Louis, Francia, Paris.

McCullagh, P. (1994), ‘Exponential mixtures and quadratic exponential families’, Biometrika 81(4), 721–729.

Mendenhall, W. & Hader, R. J. (1958), ‘Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data’, Biometrika 45(3-4), 504–520.

Norstrom, J. G. (1996), ‘The use of precautionary loss functions in risk analysis’, Reliability, IEEE Transactions on 45(3), 400–403.

Rao, G. S. (2012), ‘Estimation of Reliability in Multicomponent Stress-strength Based on Generalized Exponential Distribution’, Revista Colombiana de Estadística 35(1), 67–76.

Raqab, M. M. & Ahsanullah, M. (2001), ‘Estimation of the location and scale parameters of generalized exponential distribution based on order statistics’, Journal of Statistical Computation and Simulation 69(2), 109–123.

Romeu, L. J. (2004), ‘Censored data’, Strategic Arms Reduction Treaty 11(3), 1–8.

Saleem, M. & Aslam, M. (2009), ‘Bayesian analysis of the two component mixture of the rayleigh distribution assuming the uniform and the jeffreys priors’, Journal of Applied Statistical Science 16(4), 493–502.

Saleem, M., Aslam, M. & Economou, P. (2010), ‘On the Bayesian analysis of the mixture of power function distribution using the complete and the censored sample’, Journal of Applied Statistics 37(1), 25–40.

Sinha, S. K. (1998), Bayesian Estimation, New Age International (P) Limited, New Delhi.

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Tahir, M., & Aslam, M. (2015). Bayesian Analysis of the 3-Component Mixture of Exponential Distribution Assuming the Non-Informative Priors. Revista Colombiana De Estadística, 38(2), 431-452. https://doi.org/10.15446/rce.v38n2.51670

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