Research Article
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Year 2016, Volume: 45 Issue: 2, 609 - 628, 01.04.2016

Abstract

References

  • Abu-Taleb, A. A., Smadi, M. M. and Alawneh, A. J. Bayes estimation of the lifetime parameters for the Exponential distribution, Journal of Mathematics and Statistics 3(3), 106-108, 2007.
  • Al-Awadhi, S. A. and Gartwaite, P. H. An elicitation method for multivariate normal distributions, Communication in Statistics, Part A-Theory and Methods 27, 1123-1142, 1998.
  • Ali, M. M., Woo, J. and Nadarajah, S. Bayes estimators of the Exponential distribution, Journal of Statistics and Management Systems 8(1), 53-58, 2005.
  • Aslam, M. An Application of Prior Predictive Distribution to Elicit the Prior Density, Journal of Statistical Theory and Applications 2, 70-83, 2003.
  • Bayes, T. An essay towards solving a problem in the doctrine of chances, Philosophical Transactions of the Royal Society of London 53, 370-418, 1763.
  • Barger, K. J. Mixtures of Exponential distributions to describe the distribution of Poisson means in estimating the number of unobserved classes, M. Sc. Thesis, Cornell University, 2006.
  • Berger, J. O. Statistical decision theory and Bayesian analysis(New York: Springer-Verlag, 1985).
  • Bernardo, J. M. Reference posterior distributions for Bayesian inference, Journal of the Royal Statistical Society, Series B (Methodological), 41(2), 113-147, 1979. 624
  • Birch, R. and Bartolucci, A. A. Determination of the hyperparameters of a prior probability model in survival analysis, Computer Programs in Biomedicine 17, 89-94, 1983.
  • Chaloner, K. M. and Duncan, G. T. Assessment of beta prior distribution: PM elicitation, The Statistician 32, 174-180, 1983.
  • Davis, D. J. An analysis of some failure data, Journal of the American Statistical Association 47, 113-150, 1952.
  • DeGroot, M. H. Optimal statistical decision(McGraw-Hill, 2005).
  • de Laplace, P. S. Theorie analytique des probabilities ( Paris: Gautheir-Villars, 1820).
  • Everitt, B. and Hand, D. J. Finite mixture distribution (New York: Chapman and Hall, 1981).
  • Gavasakar, U. A comparison of two elicitation methods for a prior distribution for a binomial parameter, Management Science 34, 784-790, 1988.
  • Geisser, S. On prior distributions for binary trials, The American Statistician 38(4), 244- 247, 1984.
  • Gijbels, I. Censored data, Wiley Interdisciplinary Reviews: Computational Statistics 2(2), 178-188, 2010.
  • Hahn, E. D. Re-examining informative prior elicitation through the lens of Markovchain Monte Carlo methods, Journal of the Royal Statistical Society 169, Series A, 37-48, 2006.
  • Hebert, J. L. and Scariano, S. M. Comparing location estimators for Exponential mixtures under Pitman’s measure of closeness, Communications in Statistics- Theory and Methods 33(1), 29-46, 2005.
  • Jeffreys, H. An invariant form for the prior probability in estimation problems, Proceeding of the Royal Society of London, Series A, Mathematical and Physical Sciences 186(1007), 453-461, 1946.
  • Jeffreys, H. Theory of Probability(Oxford, UK: Claredon Press, 1961).
  • Kadane, J. B., Dickey, J. M., Winkler, R. L., Smith, W. and Peter, S. C. Interactive elicitation of opinion for a normal linear model, Journal of the American Statistical Association 75, 845-854, 1980.
  • Kalbfleisch, J. D. and Prentice, R. L. The Statistical analysis of failure time data(New York: John Wiley & Sons, 2011).
  • Kazmi, S. M. A., Aslam, M. and Ali, S. 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, 2012.
  • Legendre, A. M. Nouvelles methodes pour la determination des orbites des cometes: Appendice sur la Methode des Moindres Carres (Paris: Gautheir-Villars, 1806)
  • Li, L. A. Decomposition theorems, conditional probability, and finite mixtures distributions, Thesis, State University, New York, Albany, 1983.
  • Li, L. A. and Sedransk, N. Inference about the presence of a mixture, Technical Report, State University, New York, Albany, 1982.
  • Li, L. A. and Sedransk, N. Mixtures of distributions: A topological approach, The annals of Statistics 16(4), 1623-1634, 1988.
  • McCullagh, P. Exponential mixtures and quadratic Exponential families, Biometrika 81(4), 721-729, 1994.
  • Mendenhall, W. and Hader, R. J. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data, Biometrika 45(3-4), 504-520, 1958.
  • Norstrom, J. G. The use of precautionary loss function in risk analysis, Reliability, IEEE Transactions on 45(3), 400-403, 1996.
  • Raqab, M. M. and Ahsanullah, M. Estimation of the location and scale parameters of generalized Exponential distribution based on order statistic, Journal of Statistical Computation and Simulation 69(2), 109-123, 2001.
  • Romeu, L. J. Censored data, Strategic Arms Reduction Treaty 11(3), 1-8, 2004.
  • Saleem. M. Bayesian analysis of mixture distributions. Ph. D. Dissertation, Dept. of Statistics, Quaid-i-Azam University, Islamabad, Pakistan, 2010.
  • Saleem, M. and Aslam, M. On prior selection for the mixture of Raleigh distribution using predictive intervals, Pakistan Journal of Statistics 24(1), 21-35, 2008.
  • Saleem, M. and Aslam, M. 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, 2009.
  • Saleem, M., Aslam, M. and Economus, P. On the Bayesian analysis of the mixture of Power distribution using the complete and censored sample, Journal of Applied Statistics 37(1), 25-40, 2010.
  • Sinha, S. K. Bayesian estimation (New Delhi: New Age International (p) limited, Publisher, 1998).
  • Strelec, L. and Stehlk, M. On simulation of exact tests in Rayleigh and Normal families, AIP (American Institute of Physics) Conference Proceedings 1479, 2012.

On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions

Year 2016, Volume: 45 Issue: 2, 609 - 628, 01.04.2016

Abstract

The memory-less property of the Exponential distribution is a strong reason of its use for testing lifetimes of objects in many lifetime modeling applications. Also, mixture models have extensively been used in survival analysis and reliability studies. This article focuses on the Bayesian analysis of the 3-component mixture of Exponential distributions under type-I right censoring scheme. Taking different noninformative and informative priors, Bayes estimators and posterior risks for the unknown parameters (parameters of component distributions and mixing proportions) are derived under squared error loss function, precautionary loss function and DeGroot loss function. The elicitation of the hyperparameters is also done using prior predictive distribution.The Bayes estimators and posterior risks are looked at as a function of the test termination time. Some important properties and comparisons of the Bayes estimates are presented. Simulated results and real data example are also given to illustrate the study.

References

  • Abu-Taleb, A. A., Smadi, M. M. and Alawneh, A. J. Bayes estimation of the lifetime parameters for the Exponential distribution, Journal of Mathematics and Statistics 3(3), 106-108, 2007.
  • Al-Awadhi, S. A. and Gartwaite, P. H. An elicitation method for multivariate normal distributions, Communication in Statistics, Part A-Theory and Methods 27, 1123-1142, 1998.
  • Ali, M. M., Woo, J. and Nadarajah, S. Bayes estimators of the Exponential distribution, Journal of Statistics and Management Systems 8(1), 53-58, 2005.
  • Aslam, M. An Application of Prior Predictive Distribution to Elicit the Prior Density, Journal of Statistical Theory and Applications 2, 70-83, 2003.
  • Bayes, T. An essay towards solving a problem in the doctrine of chances, Philosophical Transactions of the Royal Society of London 53, 370-418, 1763.
  • Barger, K. J. Mixtures of Exponential distributions to describe the distribution of Poisson means in estimating the number of unobserved classes, M. Sc. Thesis, Cornell University, 2006.
  • Berger, J. O. Statistical decision theory and Bayesian analysis(New York: Springer-Verlag, 1985).
  • Bernardo, J. M. Reference posterior distributions for Bayesian inference, Journal of the Royal Statistical Society, Series B (Methodological), 41(2), 113-147, 1979. 624
  • Birch, R. and Bartolucci, A. A. Determination of the hyperparameters of a prior probability model in survival analysis, Computer Programs in Biomedicine 17, 89-94, 1983.
  • Chaloner, K. M. and Duncan, G. T. Assessment of beta prior distribution: PM elicitation, The Statistician 32, 174-180, 1983.
  • Davis, D. J. An analysis of some failure data, Journal of the American Statistical Association 47, 113-150, 1952.
  • DeGroot, M. H. Optimal statistical decision(McGraw-Hill, 2005).
  • de Laplace, P. S. Theorie analytique des probabilities ( Paris: Gautheir-Villars, 1820).
  • Everitt, B. and Hand, D. J. Finite mixture distribution (New York: Chapman and Hall, 1981).
  • Gavasakar, U. A comparison of two elicitation methods for a prior distribution for a binomial parameter, Management Science 34, 784-790, 1988.
  • Geisser, S. On prior distributions for binary trials, The American Statistician 38(4), 244- 247, 1984.
  • Gijbels, I. Censored data, Wiley Interdisciplinary Reviews: Computational Statistics 2(2), 178-188, 2010.
  • Hahn, E. D. Re-examining informative prior elicitation through the lens of Markovchain Monte Carlo methods, Journal of the Royal Statistical Society 169, Series A, 37-48, 2006.
  • Hebert, J. L. and Scariano, S. M. Comparing location estimators for Exponential mixtures under Pitman’s measure of closeness, Communications in Statistics- Theory and Methods 33(1), 29-46, 2005.
  • Jeffreys, H. An invariant form for the prior probability in estimation problems, Proceeding of the Royal Society of London, Series A, Mathematical and Physical Sciences 186(1007), 453-461, 1946.
  • Jeffreys, H. Theory of Probability(Oxford, UK: Claredon Press, 1961).
  • Kadane, J. B., Dickey, J. M., Winkler, R. L., Smith, W. and Peter, S. C. Interactive elicitation of opinion for a normal linear model, Journal of the American Statistical Association 75, 845-854, 1980.
  • Kalbfleisch, J. D. and Prentice, R. L. The Statistical analysis of failure time data(New York: John Wiley & Sons, 2011).
  • Kazmi, S. M. A., Aslam, M. and Ali, S. 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, 2012.
  • Legendre, A. M. Nouvelles methodes pour la determination des orbites des cometes: Appendice sur la Methode des Moindres Carres (Paris: Gautheir-Villars, 1806)
  • Li, L. A. Decomposition theorems, conditional probability, and finite mixtures distributions, Thesis, State University, New York, Albany, 1983.
  • Li, L. A. and Sedransk, N. Inference about the presence of a mixture, Technical Report, State University, New York, Albany, 1982.
  • Li, L. A. and Sedransk, N. Mixtures of distributions: A topological approach, The annals of Statistics 16(4), 1623-1634, 1988.
  • McCullagh, P. Exponential mixtures and quadratic Exponential families, Biometrika 81(4), 721-729, 1994.
  • Mendenhall, W. and Hader, R. J. Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data, Biometrika 45(3-4), 504-520, 1958.
  • Norstrom, J. G. The use of precautionary loss function in risk analysis, Reliability, IEEE Transactions on 45(3), 400-403, 1996.
  • Raqab, M. M. and Ahsanullah, M. Estimation of the location and scale parameters of generalized Exponential distribution based on order statistic, Journal of Statistical Computation and Simulation 69(2), 109-123, 2001.
  • Romeu, L. J. Censored data, Strategic Arms Reduction Treaty 11(3), 1-8, 2004.
  • Saleem. M. Bayesian analysis of mixture distributions. Ph. D. Dissertation, Dept. of Statistics, Quaid-i-Azam University, Islamabad, Pakistan, 2010.
  • Saleem, M. and Aslam, M. On prior selection for the mixture of Raleigh distribution using predictive intervals, Pakistan Journal of Statistics 24(1), 21-35, 2008.
  • Saleem, M. and Aslam, M. 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, 2009.
  • Saleem, M., Aslam, M. and Economus, P. On the Bayesian analysis of the mixture of Power distribution using the complete and censored sample, Journal of Applied Statistics 37(1), 25-40, 2010.
  • Sinha, S. K. Bayesian estimation (New Delhi: New Age International (p) limited, Publisher, 1998).
  • Strelec, L. and Stehlk, M. On simulation of exact tests in Rayleigh and Normal families, AIP (American Institute of Physics) Conference Proceedings 1479, 2012.
There are 39 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Muhammad Tahir This is me

Muhammad Aslam This is me

Zawar Hussain This is me

Publication Date April 1, 2016
Published in Issue Year 2016 Volume: 45 Issue: 2

Cite

APA Tahir, M., Aslam, M., & Hussain, Z. (2016). On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions. Hacettepe Journal of Mathematics and Statistics, 45(2), 609-628.
AMA Tahir M, Aslam M, Hussain Z. On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions. Hacettepe Journal of Mathematics and Statistics. April 2016;45(2):609-628.
Chicago Tahir, Muhammad, Muhammad Aslam, and Zawar Hussain. “On the Bayesian Analysis of 3-Component Mixture of Exponential Distributions under Different Loss Functions”. Hacettepe Journal of Mathematics and Statistics 45, no. 2 (April 2016): 609-28.
EndNote Tahir M, Aslam M, Hussain Z (April 1, 2016) On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions. Hacettepe Journal of Mathematics and Statistics 45 2 609–628.
IEEE M. Tahir, M. Aslam, and Z. Hussain, “On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions”, Hacettepe Journal of Mathematics and Statistics, vol. 45, no. 2, pp. 609–628, 2016.
ISNAD Tahir, Muhammad et al. “On the Bayesian Analysis of 3-Component Mixture of Exponential Distributions under Different Loss Functions”. Hacettepe Journal of Mathematics and Statistics 45/2 (April 2016), 609-628.
JAMA Tahir M, Aslam M, Hussain Z. On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions. Hacettepe Journal of Mathematics and Statistics. 2016;45:609–628.
MLA Tahir, Muhammad et al. “On the Bayesian Analysis of 3-Component Mixture of Exponential Distributions under Different Loss Functions”. Hacettepe Journal of Mathematics and Statistics, vol. 45, no. 2, 2016, pp. 609-28.
Vancouver Tahir M, Aslam M, Hussain Z. On the Bayesian analysis of 3-component mixture of exponential distributions under different loss functions. Hacettepe Journal of Mathematics and Statistics. 2016;45(2):609-28.