| Peer-Reviewed

Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution

Received: 17 August 2013     Published: 30 August 2013
Views:       Downloads:
Abstract

In this paper, based on a new type of censoring scheme called a progressive first-failure censored, the maximum likelihood (ML) and the Bayes estimators for the two unknown parameters of the Generalized Pareto (GP) distribution are derived. This type of censoring contains as special cases various types of censoring schemes used in the literature. A Bayesian approach using Markov Chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are developed. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are also proposed. The approximate Bayes estimators have been obtained under the assumptions of informative and non-informative priors. A numerical example is provided to illustrate the proposed methods. Finally, the maximum likelihood and different Bayes estimators are compared via a Monte Carlo simulation study.

Published in American Journal of Theoretical and Applied Statistics (Volume 2, Issue 5)
DOI 10.11648/j.ajtas.20130205.13
Page(s) 128-141
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), 2013. Published by Science Publishing Group

Keywords

Generalized Pareto Distribution, Progressive First-Failure Censored Sample, Gibbs and Metropolis Sampler, Bayesian and Non-Bayesian Estimations, Bootstrap Methods

References
[1] Abd Ellah, A.H. (2003). Bayesian one sample prediction bounds for the Lomax distribution. Indian Journal of Pure and Applied Mathematics, 34, 101-109.
[2] 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.
[3] Arnold, B.C. (1983). Pareto distributions. In: Statistical distributions in scientific Work. International Co-operative Publishing House, Burtonsville, MD.
[4] Balakrishnan, N. and Sandhu, R.A. (1995). A simple simulation algorithm for generating progressively type-II censored samples. American Statistics, 49, 229-230.
[5] Bryson, M.C. (1974). Heavy-tailed distributions: Properties and tests. Technometrics, 16(1), 61--68.
[6] Chahkandi, M. and Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics and Data Analysis, 53(12), 4433--4440.
[7] Efron, B. (1982). The Bootstrap and other resampling plans, In: CBMS-NSF Regional Conference Seriesin Applied Mathematics, SIAM, Philadelphia, PA.
[8] Habibullah, M. and Ahsanullah, M. (2000). Estimation of parameters of a Pareto distribution by generalized order statistics. Communication in Statistics-Theory and Methods, 29, 1597-1609.
[9] Hall, P. (1988). Theoretical comparison of Bootstrap confidence intervals. Annals of Statistics, 16, 927-953.
[10] Johnson, L.G. (1964). Theory and technique of variation research, Elsevier, Amsterdam.
[11] Jun, C.-H.,Balamurali, S. and Lee, S.-H. (2006) Variables sampling plans for Weibull distributed lifetimes under sudden death testing. IEEE Transaction on Reliability, 55, 53-58.
[12] Lee, W.-C., Wu, L.-W. and Yu, H.-Y. (2007). Statistical inference about the shape parameter of the Bathtub-Shaped distribution under the failure-censored sampling plan. International Journal of Information Management Science, 18, 157-172.
[13] Lomax, K.S. (1954). Business failure: Another example of the analysis of the failure data. JASA, 49, 847-852.
[14] Marshall, A.W. and Olkin, I. (2007). Life distributions structure of nonparametric, semiparametric, and parametric families, Springer, New York, NY.
[15] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemistry and Physics, 21, 1087--1091.
[16] Rezaei, R., Tahmasbi, R. and Mahmoodi, M. (2010). Estimation of P[Y
[17] Robert, C.P. and Casella, G. (2004). Monte Carlo statistical methods, Second edition. Springer: New York.
[18] Soliman, A.A., Abd Ellah, A.H., Abou-Elheggag, N.A. and Modhesh, A.A. (2012a). Estimation from Burr type XII distribution using progressive first-failure censored data. Journal of Statistical Computation and Simulation, iFirst, 1--21.
[19] Soliman, A.A., Abd Ellah, A.H., Abou-Elheggag, N.A. and Abd-Elmougod, G.A. (2012b). Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data. Computational Statistics and Data Analysis, 56, 2471--2485.
[20] Soliman, A.A., Abd Ellah, A.H., Abou-Elheggag, N.A. and Modhesh, A.A. (2011a). Bayesian inference and prediction of Burr type XII distribution for progressive first-failure censored sampling, Intelligent Information Management, 3, 175-185.
[21] Soliman, A.A., Abd Ellah, A.H., Abou-Elheggag, N.A. and Abd-Elmougod, G.A. ( 2011b). Simulation-based approach to the study of coefficient of variation of Gompertz distribution under progressive first-failure censoring. Indian Journal of Pure and Applied Mathematics, 42(5), 335-356.
[22] Upadhyay, S.K. and Peshwani, M. (2003). Choice between Weibull and Lognormal models: A simulation based Bayesian study. Communication in Statistics-Theory and Methods, 32, 381-405.
[23] Wu, J.-W., Hung, W.-L. and Tsai, C.-H. (2003). Estimation of the parameters of the Gompertz distribution under the first-failure-censored sampling plan. Statistics, 37(6), 517-525.
[24] Wu, J.-W. and Yu, H.-Y. (2005). Statistical inference about the shape parameter of the Burr type XII distribution under the failure-censored sampling plan. Applied Mathematics and Computation, 163, 443-482.
[25] Wu, J.-W., Ouyang, T.-R. and Yu, L.-Y. (2001). Limited failure-censored life test for the Weibull distribution, IEEE Transaction on Reliability, 50, 107-111.
[26] Wu, S.-J. and Kuş, (2009). On estimation based on progressive first-failure censored sampling. Computational Statistics and Data Analysis, 53(10), 3659-3670.
Cite This Article
  • APA Style

    Mohamed Abdul Wahab Mahmoud, Ahmed Abo-Elmagd Soliman, Ahmed Hamed Abd Ellah, Rashad Mohamed El-Sagheer. (2013). Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution. American Journal of Theoretical and Applied Statistics, 2(5), 128-141. https://doi.org/10.11648/j.ajtas.20130205.13

    Copy | Download

    ACS Style

    Mohamed Abdul Wahab Mahmoud; Ahmed Abo-Elmagd Soliman; Ahmed Hamed Abd Ellah; Rashad Mohamed El-Sagheer. Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution. Am. J. Theor. Appl. Stat. 2013, 2(5), 128-141. doi: 10.11648/j.ajtas.20130205.13

    Copy | Download

    AMA Style

    Mohamed Abdul Wahab Mahmoud, Ahmed Abo-Elmagd Soliman, Ahmed Hamed Abd Ellah, Rashad Mohamed El-Sagheer. Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution. Am J Theor Appl Stat. 2013;2(5):128-141. doi: 10.11648/j.ajtas.20130205.13

    Copy | Download

  • @article{10.11648/j.ajtas.20130205.13,
      author = {Mohamed Abdul Wahab Mahmoud and Ahmed Abo-Elmagd Soliman and Ahmed Hamed Abd Ellah and Rashad Mohamed El-Sagheer},
      title = {Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {2},
      number = {5},
      pages = {128-141},
      doi = {10.11648/j.ajtas.20130205.13},
      url = {https://doi.org/10.11648/j.ajtas.20130205.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20130205.13},
      abstract = {In this paper, based on a new type of censoring scheme called a progressive first-failure censored, the maximum likelihood (ML) and the Bayes estimators for the two unknown parameters of the Generalized Pareto (GP) distribution are derived. This type of censoring contains as special cases various types of censoring schemes used in the literature. A Bayesian approach using Markov Chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are developed. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are also proposed. The approximate Bayes estimators have been obtained under the assumptions of informative and non-informative priors. A numerical example is provided to illustrate the proposed methods. Finally, the maximum likelihood and different Bayes estimators are compared via a Monte Carlo simulation study.},
     year = {2013}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution
    AU  - Mohamed Abdul Wahab Mahmoud
    AU  - Ahmed Abo-Elmagd Soliman
    AU  - Ahmed Hamed Abd Ellah
    AU  - Rashad Mohamed El-Sagheer
    Y1  - 2013/08/30
    PY  - 2013
    N1  - https://doi.org/10.11648/j.ajtas.20130205.13
    DO  - 10.11648/j.ajtas.20130205.13
    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  - 128
    EP  - 141
    PB  - Science Publishing Group
    SN  - 2326-9006
    UR  - https://doi.org/10.11648/j.ajtas.20130205.13
    AB  - In this paper, based on a new type of censoring scheme called a progressive first-failure censored, the maximum likelihood (ML) and the Bayes estimators for the two unknown parameters of the Generalized Pareto (GP) distribution are derived. This type of censoring contains as special cases various types of censoring schemes used in the literature. A Bayesian approach using Markov Chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are developed. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are also proposed. The approximate Bayes estimators have been obtained under the assumptions of informative and non-informative priors. A numerical example is provided to illustrate the proposed methods. Finally, the maximum likelihood and different Bayes estimators are compared via a Monte Carlo simulation study.
    VL  - 2
    IS  - 5
    ER  - 

    Copy | Download

Author Information
  • Mathematics Department, Faculty of Science, A1-Azhar University, Nasr-City 11884, Cairo, Egypt

  • Mathematics Department, Faculty of Science, Islamic University, Madinh, Saudi Arabia

  • Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt

  • Mathematics Department, Faculty of Science, A1-Azhar University, Nasr-City 11884, Cairo, Egypt

  • Sections