Estimation of Generalized Pareto under an Adaptive Type-II Progressive Censoring

Mohamed A. W. Mahmoud, Ahmed A. Soliman, Ahmed H. Abd Ellah, Rashad M. El-Sagheer

Faculty of Science, Islamic University, Madinah, Saudi Arabia.
Mathematics Department, Faculty of Science, A1-Azhar University, Nasr-City, Cairo, Egypt.
Mathematics Department, Sohag University, Sohag, Egypt.

DOI: 10.4236/iim.2013.53008                

Abstract

In this paper, based on a new type of censoring scheme called an adaptive type-II progressive censoring scheme introduce by Ng et al. [1], Naval Research Logistics is considered. Based on this type of censoring the maximum likelihood estimation (MLE), Bayes estimation, and parametric bootstrap method are used for estimating the unknown parameters. Also, we propose to apply Markov chain Monte Carlo (MCMC) technique to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. Point estimation and confidence intervals based on maximum likelihood and bootstrap method are also proposed. The approximate Bayes estimators obtained under the assumptions of non-informative priors, are compared with the maximum likelihood estimators. Numerical examples using real data set are presented to illustrate the methods of inference developed here. Finally, the maximum likelihood, bootstrap and the different Bayes estimates are compared via a Monte Carlo simulation study.

Keywords

Generalized Pareto (GP) Distribution; An Adaptive Type-II Progressive Censoring Scheme; Bayesian and Non-Bayesian Estimations; Gibbs and Metropolis Sampler; Bootstrap

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. K. T. Ng, D. Kudu and P. S. Chan, “Statistical Analysis of Exponential Lifetimes under an Adaptive Type-II Progressive Censoring Scheme,” Naval Research Logistics, Vol. 56, No. 8, 2010, pp. 687-698. doi:10.1002/nav.20371
[2]	N. Bal Krishnan, “Progressive Censoring Methodology: An Appraisal,” Test, Vol. 16, No. 2, 2007, pp. 211-296. doi:10.1007/s11749-007-0061-y
[3]	N. Bal Krishnan and R. Aggarwala, “Progressive Censoring: Theory, Methods, and Applications,” Birkhauser, Boston, Berlin, 2000.
[4]	K.S. Lomax, “Business Failure: Another Example of the Analysis of the Failure Data,” Journal of the American Statistical Association, Vol. 49, No. 268, 1954, pp. 847-852. doi:10.1080/01621459.1954.10501239
[5]	M. Habibullah and M. Ahsanullah, “Estimation of Parameters of a Pareto distribution by Generalized Order Statstics,” Communication in Statistics, Theory and Methods, Vol. 29, No. 7, 2000, pp. 1597-1609. doi:10.1080/03610920008832567
[6]	S. K. Upadhyay and M. Peshwani, “Choice between Weibull and Lognormal Models: A Simulation Based Bayesian Study,” Communication in Statistics, Theory and Methods, Vol. 32, No. 2, 2003, pp. 381-405. doi:10.1081/STA-120018191
[7]	A. H. Abd Ellah, “Bayesian One Sample Prediction Bounds for the Lomax Distribution,” Indian Journal Pure and Applied Mathematics, Vol. 34, No. 1, 2003, pp. 101-109.
[8]	A. H. Abd Ellah, “Comparison of Estimates Using Record Statstics from Lomax Model: Bayesian and Non Bayesian Approaches,” Journal of Statistical Research and Training Center, Vol. 3, No. 2, 2006, pp. 139-158.
[9]	E. Cramer and G. Iliopoulos, “Adaptive Progressive Type-II Censoring,” Test, Vol. 19, No. 2, 2010, pp. 342-358. doi:10.1007/s11749-009-0167-5
[10]	H. A. David and H. N. Nagaraja, “Order Statistics,” 3rd Edition, Wiley, New York, 2003. doi:10.1002/0471722162
[11]	H. K. T. Ng and P. S. Chan, “Comments on Progressive Censoring Methodology: An Appraisal,” Test, Vol. 16, No. 2, 2007, pp. 287-289. doi:10.1007/s11749-007-0071-9
[12]	B. Efron, “The Jackknife, the Bootstrap and Other Resampling Plans,” CBMS-NSF Regional Conference Seriesin, Applied Mathematics, SIAM, Philadelphia, Vol. 38, 1982.
[13]	C. P. Robert and G. Casella, “Monte Carlo Statistical Methods,” 2nd Edition, Springer, New York, 2004. doi:10.1007/978-1-4757-4145-2
[14]	S. Rezaei, R. Tahmasbi and M. Mahmoodi, “Estimation of P[Y
[15]	N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, “Equations of State Calculations by fast Computing Machines,” Journal Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1091. doi:10.1063/1.1699114
[16]	W. K. Hastings, “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, No. 1, 1970, pp. 97-109. doi:10.1093/biomet/57.1.97
[17]	A. A. Soliman, A. H. Abd-Ellah, N. A. Abou-Elheggag and E. A. Ahmed, “Reliability Estimation in Stress-Strength Models: An MCMC Approach,” Statistics, 2011, pp. 1-14.doi:10.1080/02331888.2011.637629
[18]	A. A. Soliman, A. H. Abd-Ellah, N. A. Abou-Elheggag and E. A. Ahmed, “Modified Weibull Model: A Bayes Study Using MCMC Approach Based on Progressive Censoring Data,” Reliability Engineering and System Safety, Vol. 100, No. 2, 2012, pp. 48-57. doi:10.1016/j.ress.2011.12.013
[19]	W. B. Nelson, “Applied Life Data Analysis,” Wiley, New York, 1982. doi:10.1002/0471725234