Bayesian Estimation via numerical approximations under Progressive Type II Censoring

Ranjita Pandey¹ , Neera Kumari²

Section:Research Paper, Product Type: Isroset-Journal
Vol.5 , Issue.6 , pp.362-379, Dec-2018

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.362379

Online published on Dec 31, 2018

Copyright © Ranjita Pandey, Neera Kumari . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
Classical and Bayesian estimation of the unknown parametric functions for power generalized Weibull distribution under progressive Type II censoring scheme are undertaken in the present paper. Newton Raphson iterative procedure is used for computation of maximum likelihood estimates which are not obtained in closed form. Asymptotic and bootstrap confidence intervals are also obtained. Squared error and general entropy loss functions are considered for Bayes estimation under the assumption of two independent gamma priors. The approximate Bayes estimates are obtained using Tierney-Kadane approximation. Alternatively, Meteropolis Hastings algorithm is run under Gibbs sampler environment to generate Bayes etimates. Computed Bayes estimates are compared with the classical maximum likelihood estimates based a simulated data and a real data set.

Key-Words / Index Term :
Type II progressive right censoring scheme, Boot-p and Boot-t intervals, Tierney and Kadane method, Markov Chain Monte Carlo.

References :
[1] W. Q. Meeker, L. A. Escobar, “Statistical Methods for Reliability Data”,Wiley, New York. 1998.
[2] A. C. Cohen, “Progressively Censored Samples in Life Testing”. Technometrics, Vol. 5, pp. 327–339. 1963.
[3] N. Balakrishnan, R. Aggarwalla, “Progressive Censoring: Theory, Methods and Applications”. Birkhauser, Boston, USA. 2000.
[4] V. Bagdonavicius, M. Nikulin, “Accelerated Life Models”. Chapman and Hall/CRC, Boca Raton, Florida, 2002.
[5] V. Voinov, N. Pya, N. Shapakov, Y.Voinov, “Goodness-of-fit tests for the power generalized weibull probability distribution”. Communications in Statistics-Simulation and Computation, Vol. 42 , pp.1003-1012, 2013.
[6] N. Nikulin, F. Haghighi, “On the power generalized Weibull family: model for cancer censored data”, Metron,Vol. LXVII, pp.75-86, 2009.
[7] R. Pandey , N. Kumari, “Bayesian Analysis of Power Generalized Weibull Distribution”, International Journal of Applied and Computational Mathematics, Vol. 4, Issue 6, pp. 141
[8] R. Calabria, G.Pulcini, “An Engineering Approach to Bayes Estimation for the Weibull Distribution”, Microelectronics Reliability, Vol. 34, Issue5, pp. 789-802, 1994.
[9] L. Tierney, J. B. Kadane, “Accurate Approximations for Posterior Moments and Marginal Densities”. Journal of American Statistical Association, Vol. 81, Issue 393, PP. 82-86.1986.
[10] W. K. Pang, S.H. Hou, W. T. Yu, “ On a Proper way to Select Population Failure Distribution and a Stochastic Optimization Method in Parameter Estimation”, European Journal of Operation Research, Vol. 177, pp. 604–611, 2007.
[11] A. A. Soliman, A. H. Abd-Ellah, N. A. Abou-Elheggag, E. A. Ahmed, “Modified Weibull model: A Bayes study using MCMC Approach based on Progressive Censoring Data”. Reliability Engineering and System Safety, Elsevier, Vol. 100, pp. 48-57, 2012.
[12] W. H. Greene, “Econometric Analysis”. 4th ed., International ed., London: Prentice-Hall International (UK), 2000.
[13] A. Agresti, “Categorical Data Analysis” Second Edition. John Wiley & Sons, 2002.
[14] B. Efron, “Bootstrap Methods: Another Look at the Jackknife”, Annals of Statistics, Vol.7, pp. 1-26, 1973.
[15] B. Efron, “The Jackknife, the Bootstrap and other Resampling Plans. CBMS - NSF Regional Conference Series in Applied Mathematics”. No. 38, Philadelphia (PA): SIAM, 1982.
[16] C. Robert, G. Casella, “Introducing Monte Carlo Methods with R” Springer, 2009.
[17] M. H. Chen, Q. M. Shao, J.G. Ibrahim, “Monte Carlo Methods in Bayesian Computation”. Springer Series in Statistics. 2012.
[18] M. H. Chen, Q. M. Shao, “Monte Carlo estimation of Bayesian credible and HPD intervals”. Journal of Computational and Graphical Statistics, Vol. 8, pp. 69 – 92, 1999.
[19] R. S. Chhikara, J. L. Folks, “The Inverse Gaussian Distribution as a Lifetime Model”, Technometrics, Vol. 19, pp. 461-468, 1977.