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

Rashad Mohamed El-Sagheer

doi:doi:10.11648/j.ajtas.20150401.15

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

Rashad Mohamed El-Sagheer

Published in American Journal of Theoretical and Applied Statistics (Volume 4, Issue 1)

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

[1]	Arnold, B. C., 1983. Pareto distributions. In: Statistical Distributions i Scientific Work, International Co-operative Publishing House, Burtonsville, MD.
[2]	Arnold, B. C., Balakrishnan, N. & Nagaraja, H. N.,1998. Records, Wiley, New York.
[3]	Resnick, S. I., 1987. Extreme values, regular variation, and point processes, Springer-Verlag,New York.
[4]	Raqab, M. Z. & Ahsanulla, M., 2001. Estimation of the location and scale parameters of the generalized exponential distribution based on order Statistics. Journal of Statistical Computation & Simulation, 69: 109- 124.
[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.
[7]	Ahsanullah, M., 1995. Introduction to record statistics, NOVA Science Publishers Inc., Hunting ton, New York.
[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.
[15]	Efron, B., 1982. The bootstrap and other resampling plans, In: CBMS-NSF Regional Conference Seriesin Applied Mathematics, SIAM, Philadelphia, PA, 1982.
[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.
[18]	Smith, A. & Roberrs, G., 1993. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of Reliability & Statistical Society, 55: 3-23.
[19]	Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A.H. & Teller, E., 1953. Equations of state calculations by fast computing machine. Journal of Chemistry Physics, 21: 1087-1092.
[20]	Hastings, W. K., 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika,57: 97-101.
[21]	Robert, C. P. & Casella, G., 2004. Monte Carlo statistical methods, second edition, Springer, New York.
[22]	Rezaei, S., Tahmasbi, R. &Mahmoodi, M., 2010. Estimation of P[Y

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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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TY  - JOUR
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
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150401.15
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
SP  - 26
EP  - 32
PB  - Science Publishing Group
SN  - 2326-9006
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
ER  -

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Author Information

Rashad Mohamed El-Sagheer

Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt

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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}
}

Copy | Download

TY  - JOUR
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
PY  - 2015
N1  - https://doi.org/10.11648/j.ajtas.20150401.15
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
SP  - 26
EP  - 32
PB  - Science Publishing Group
SN  - 2326-9006
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
ER  -

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