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Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach

Received: 10 January 2015     Accepted: 13 January 2015     Published: 23 January 2015
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Abstract

In this paper, we consider the Bayes estimators of the unknown parameters of the exponentiated Weibull distribution (EWD) under the assumptions of gamma priors on both shape parameters. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are proposed. The Bayes estimators cannot be obtained in explicit forms. So we propose Markov chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimators. The approximate Bayes estimators obtained under the assumptions of non-informative priors are compared with the maximum likelihood estimators using Monte Carlo simulations. A numerical example is also presented for illustrative purposes.

Published in American Journal of Theoretical and Applied Statistics (Volume 4, Issue 1)
DOI 10.11648/j.ajtas.20150401.15
Page(s) 26-32
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Exponentiated Weibull Distribution (EWD), Record Values, Bootstrap Methods, Bayes Estimation, Gibbs and Metropolis Sampler

References
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[3] Resnick, S. I., 1987. Extreme values, regular variation, and point processes, Springer-Verlag,New York.
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[5] Nagaraja, H. N., 1988. Record values and related statistics-a review. Communication in Statistics Theory & Methods,17: 2223-2238.
[6] Ahsanullah, M., 1993. On the record values from univariate distributions, National Institute of Standards and Technology Journal of Research, Special Publications, 12: 1-6.
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[8] Raqab, M. Z., 2002. Inferences for generalized exponential distribution based on record Statistics. Journal of Statistical Planning &Inference, 52: 339-350.
[9] Abd Ellah, A. H., 2011. Bayesian one sample prediction bounds for the Lomax distribution. Indian Journal of Pure and Applied Mathematics, 42: 335-356.
[10] Abd Ellah, A. H., 2006. Comparison of estimates using record statistics from Lomax model: Bayesian and non Bayesian approaches. Journal of Statistical Research and Training Center, 3: 139-158.
[11] Sultan, K. S. & Balakrishnan, N., 1999. Higher order moments of record values from Rayleigh and Weibull distributions and Edgeworth approximate inference, Journal of Applied Statistical Science, 9: 193-209.
[12] Preda, V. & Panaitescu, E., 2010. Constantinescu and S. Sudradjat, Estimations and predictions using record statistics from the modified Weibull model, WSEAS Transaction on Mathematics, 9: 427-437.
[13] Mahmoud, M. A. W., Soliman, A. A., Abd Ellah, A. H. & EL-Sagheer, R. M., 2013. Markov chain Monte Carlo to study the estimation of the coefficient of variation. International Journal of Computer Applications, 77: 31-37.
[14] Mudholkar, G. S. & Srivastava, D. K., 1995. The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics, 37(4): 436-445.
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[16] Hall, P., 1988. Theoretical comparison of bootstrap confidence intervals. Annals in Statistics, 16: 927-953.
[17] Geman, S. & Geman, D., 1984. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEE Transaction on Pattern Analysis and Machine Intelligent, 12: 609-628.
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  • APA Style

    Rashad Mohamed El-Sagheer. (2015). Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach. American Journal of Theoretical and Applied Statistics, 4(1), 26-32. https://doi.org/10.11648/j.ajtas.20150401.15

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    ACS Style

    Rashad Mohamed El-Sagheer. Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach. Am. J. Theor. Appl. Stat. 2015, 4(1), 26-32. doi: 10.11648/j.ajtas.20150401.15

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    AMA Style

    Rashad Mohamed El-Sagheer. Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach. Am J Theor Appl Stat. 2015;4(1):26-32. doi: 10.11648/j.ajtas.20150401.15

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  • @article{10.11648/j.ajtas.20150401.15,
      author = {Rashad Mohamed El-Sagheer},
      title = {Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {4},
      number = {1},
      pages = {26-32},
      doi = {10.11648/j.ajtas.20150401.15},
      url = {https://doi.org/10.11648/j.ajtas.20150401.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20150401.15},
      abstract = {In this paper, we consider the Bayes estimators of the unknown parameters of the exponentiated Weibull distribution (EWD) under the assumptions of gamma priors on both shape parameters. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are proposed. The Bayes estimators cannot be obtained in explicit forms. So we propose Markov chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimators. The approximate Bayes estimators obtained under the assumptions of non-informative priors are compared with the maximum likelihood estimators using Monte Carlo simulations. A numerical example is also presented for illustrative purposes.},
     year = {2015}
    }
    

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    T1  - Bayesian Estimation Based on Record Values from Exponentiated Weibull Distribution: An Markov Chain Monte Carlo Approach
    AU  - Rashad Mohamed El-Sagheer
    Y1  - 2015/01/23
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    DO  - 10.11648/j.ajtas.20150401.15
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
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    UR  - https://doi.org/10.11648/j.ajtas.20150401.15
    AB  - In this paper, we consider the Bayes estimators of the unknown parameters of the exponentiated Weibull distribution (EWD) under the assumptions of gamma priors on both shape parameters. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are proposed. The Bayes estimators cannot be obtained in explicit forms. So we propose Markov chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimators. The approximate Bayes estimators obtained under the assumptions of non-informative priors are compared with the maximum likelihood estimators using Monte Carlo simulations. A numerical example is also presented for illustrative purposes.
    VL  - 4
    IS  - 1
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Author Information
  • Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt

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