Estimations of Weibull-Geometric Distribution under Progressive Type II Censoring Samples
Affiliation(s)
1 Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Taif University, Ranyah Branch, Saudi Arabia. 3 Department of Mathematics, Faculty of Science, Fayoum University, Al Fayoum, Egypt. 4 Department of Computer Science, College of Computer and IT, Taif University, Taif, Saudi Arabia.
1 Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia. 2 Department of Mathematics, Faculty of Science, Taif University, Ranyah Branch, Saudi Arabia. 3 Department of Mathematics, Faculty of Science, Fayoum University, Al Fayoum, Egypt. 4 Department of Computer Science, College of Computer and IT, Taif University, Taif, Saudi Arabia.
ABSTRACT
This paper deals with the Bayesian inferences of unknown parameters of the progressively Type II censored Weibull-geometric (WG) distribution. The Bayes estimators cannot be obtained in explicit forms of the unknown parameters under a squared error loss function. The approximate Bayes estimators will be computed using the idea of Markov Chain Monte Carlo (MCMC) method to generate from the posterior distributions. Also the point estimation and confidence intervals based on maximum likelihood and bootstrap technique are also proposed. The approximate Bayes estimators will be obtained under the assumptions of informative and non-informative priors are compared with the maximum likelihood estimators. A numerical example is provided to illustrate the proposed estimation methods here. Maximum likelihood, bootstrap and the different Bayes estimates are compared via a Monte Carlo Simulation study
KEYWORDS
Weibull-Geometric Distribution, Progressive Type II Censoring Samples, Bayesian Estimation, Maximum Likelihood Estimation, Bootstrap Confidence Intervals, Markov Chain Monte Carlo
Weibull-Geometric Distribution, Progressive Type II Censoring Samples, Bayesian Estimation, Maximum Likelihood Estimation, Bootstrap Confidence Intervals, Markov Chain Monte Carlo
Cite this paper
Elhag, A. , Ibrahim, O. , El-Sayed, M. and Abd-Elmougod, G. (2015) Estimations of Weibull-Geometric Distribution under Progressive Type II Censoring Samples. Open Journal of Statistics, 5, 721-729. doi: 10.4236/ojs.2015.57072.
Elhag, A. , Ibrahim, O. , El-Sayed, M. and Abd-Elmougod, G. (2015) Estimations of Weibull-Geometric Distribution under Progressive Type II Censoring Samples. Open Journal of Statistics, 5, 721-729. doi: 10.4236/ojs.2015.57072.
References
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[1] Green, E.J., Roesh Jr., F.A., Smith, A.F.M. and Strawderman, W.E. (1994) Bayes Estimation for the Three Parameter Weibull Distribution with Tree Diameter Data. Biometrics, 50, 254-269. [2] Adamidis, K. and Loukas, S. (1998) A Lifetime Distribution with Decreasing Failure Rate. Statistics & Probability Letters, 39, 35-42. [3] Marshall, A.W. and Olkin, I. (1997) A New Method for Adding a Parameter to a Family of Distributions with Application to the Exponential and Weibull Families. Biometrika, 84, 641-652. http://dx.doi.org/10.1093/biomet/84.3.641 [4] Adamidis, K., Dimitrakopoulou, T. and Loukas, S. (2005) On a Generalization of the Exponential-Geometric Distribution. Statistics & Probability Letters, 73, 259-269. [5] Kus, C. (2007) A New Lifetime Distribution. Computational Statistics & Data Analysis, 51, 4497-4509. http://dx.doi.org/10.1016/j.csda.2006.07.017 [6] Souza, W.A., Morais, A.L. and Cordeiro, G.M. (2010) The Weibull-Geometric Distribution. Journal of Statistical Computation and Simulation, 81, 645-657. http://dx.doi.org/10.1080/00949650903436554 [7] Barreto-Souza, W. (2011) The Weibull-Geometric Distribution. Journal of Statistical Computation and Simulation, 81, 645-657. http://dx.doi.org/10.1080/00949650903436554 [8] Hamedani, G.G. and Ahsanullah, M. (2011) Characterizations of Weibull-Geometric Distribution. Journal of Statistical Theory and Applications, 10, 581-590. [9] Balakrishnan, N. and Sandhu, R.A. (1995) A Simple Simulation Algorithm for Generating Progressively Type-II Censored Samples. The American Statistician, 49, 229-230. [10] Balakrishnan, N. and Aggarwala, R. (2000) Progressive Censoring: Theory, Methods, and Applications. Birkhauser, Boston. http://dx.doi.org/10.1007/978-1-4612-1334-5 [11] Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996) Markov Chain Monte Carlo in Practices. Chapman and Hall, London. [12] Gamerman, D. (1997) Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Chapman and Hall, London. [13] Geman, S. and Geman, D. (1984) Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Mathematical Intelligence, 6, 721-741. http://dx.doi.org/10.1109/TPAMI.1984.4767596 [14] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953) Equations of State Calculations by Fast Computing Machines. Journal Chemical Physics, 21, 1087-1091. http://dx.doi.org/10.1063/1.1699114 [15] Hastings, W.K. (1970) Monte Carlo Sampling Methods Using Markov Chains and Their Applications. Biometrika, 57, 97-109. http://dx.doi.org/10.1093/biomet/57.1.97 [16] Gelfand, A.E. and Smith, A.F.M. (1990) Sampling Based Approach to Calculating Marginal Densities. Journal of the American Statistical Association, 85, 398-409. http://dx.doi.org/10.1080/01621459.1990.10476213 [17] Efron, B. and Tibshirani, R.J. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York. http://dx.doi.org/10.1007/978-1-4899-4541-9 [18] Efron, B. (1982) The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics, Phiadelphia, 38. http://dx.doi.org/10.1137/1.9781611970319