Bayesian Estimation Based on Record Values From Exponentiated Weibull Distribution: an Markov Chain Monte Carlo Approach

Author(s)

Rashad Mohamed El-Sagheer ,

Download Full PDF Pages: 01-11 | Views: 838 | Downloads: 215 | DOI: 10.5281/zenodo.3413398

Volume 3 - October 2014 (10)

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.

Keywords

Exponentiated Weibull distribution (EWD), Record values, Bootstrap methods, Bayes estimation, Gibbs and Metropolis sampler. 

References

i.        Arnold, B. C., 1983. Pareto distributions. In: Statistical Distributions i Scientific Work, International Co-operative Publishing House, Burtonsville, MD.

ii.      Arnold, B. C., Balakrishnan, N. & Nagaraja, H. N.,1998. Records, Wiley, New York.

iii.    Resnick, S. I., 1987. Extreme values, regular variation, and point processes, Springer-Verlag,New York.

iv.     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.

v.       Nagaraja, H. N., 1988. Record values and related statistics-a review. Communication in Statistics Theory & Methods,17: 2223-2238

vi.     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.

vii.   Ahsanullah, M., 1995. Introduction to record statistics, NOVA Science Publishers Inc., Hunting ton, New York.

viii. Raqab, M. Z., 2002. Inferences for generalized exponential distribution based on record Statistics. Journal of Statistical Planning &Inference, 52: 339-350.

ix.     Abd Ellah, A. H., 2011. Bayesian one sample prediction bounds for the Lomax distribution. Indian Journal of Pure and Applied Mathematics, 42: 335-356.

x.       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.

xi.     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.

xii.   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.

xiii. Mahmoud, M. A. W., Soliman, A. A., Abd Ellah, A. H. & EL-Sagheer, R. M., 2013. Markov chain Monte Carlo to study the

xiv. Mudholkar, G. S. & Srivastava, D. K., 1995. The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics, 37(4): 436-445.

xv.   Efron, B., 1982. The bootstrap and other resampling plans, In: CBMS-NSF Regional Conference Seriesin Applied Mathematics, SIAM, Philadelphia, PA, 1982.

xvi. Hall, P., 1988. Theoretical comparison of bootstrap confidence intervals. Annals in Statistics, 16: 927-953.

xvii. 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.

xviii.  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.

xix.  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

xx.   Hastings, W. K., 1970. Monte Carlo sampling methods using Markov chains and their applications. Biometrika,57: 97-101.

xxi. Robert, C. P. & Casella, G., 2004. Monte Carlo statistical methods, second edition, Springer, New York.

xxii.   Rezaei, S., Tahmasbi, R. &Mahmoodi, M., 2010. Estimation of P[Y<X] for generalized Pareto distribution, Journal of Statistical Planning & Inference, 140: 480-494.

Cite this Article: