Bayesian estimation of Rayleigh distribution in the presence of outliers using progressive censoring

Year 2020, Volume: 49 Issue: 6, 2119 - 2133, 08.12.2020

Farzana Noor Ahthasham Sajid Maha Ghazal İmranullah Khan Mehwish Zaman İmran Baig

https://doi.org/10.15672/hujms.635684

Abstract

In this article, Maximum likelihood estimation (MLE) and Bayesian estimation for Rayleigh distribution using progressive type-II censoring in the presence of outliers is considered. Inverse Gamma prior and Jeffreys prior are used for Bayesian estimation. Squared error loss function (SELF), precautionary loss function (PLF) and K-loss function (KLF) are used for obtaining the expressions of Bayes estimators and posterior risks. Credible intervals are also derived. A simulation study is presented to discuss the behavior of Bayes estimators. Applicability of the undertaken study is highlighted using three real data sets.

Keywords

Rayleigh distribution, outliers, progressive type-II censoring, life testing experiments, prior distribution, loss function

References

• [1] T.A. Abushal, Estimation of the unknown parameters for the compound Rayleigh distribution based on progressive first-failure-censored sampling, OJS 1 (03), 161–171, 2011.
• [2] W.M. Afify, Classical estimation of mixed Rayleigh distribution in type-I progressive censored, Stat. Theory Appl. 10 (4), 619–632, 2011.
• [3] S. Ali, Mixture of the inverse Rayleigh distribution: Properties and estimation in a Bayesian framework, Appl. Math. Model. 39, 515–530, 2015.
• [4] S. Ali, On the Bayesian estimation of the weighted Lindley distribution, J. Stat. Comput. Simul. 85 (5), 855-880, 2015.
• [5] S. Ali, M. Aslam, A. Nasir and S.M.A. Kazmi, Scale parameter estimation of the Laplace model using different asymmetric loss functions, International Journal of Statistics and Probability 1 (1), 105–127, 2012.
• [6] F.J. Anscombe, Residuals, in: Time series and Statistics, 244-250, Palgrave Macmillan, London, 1990.
• [7] M. Aslam, F. Noor and S. Ali, Shifted Exponential distribution: Bayesian estimation, prediction and expected test time under progressive censoring, J. Test. Eval. 48 (2), 1576–1593, 2020.
• [8] R. Azimi and F. Yaghmaei, Bayesian estimation based on Rayleigh progressive type II censored data with Binomial removals, Journal of Quality and Reliability Engineering, 1–6, 2013.
• [9] N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications, Springer Science and Business Media, 2000.
• [10] V. Barnett, The study of outliers: Purpose and Model, J. R. Stat. Soc. Ser. C. Appl. Stat. 27 (3), 242–250, 1978.
• [11] A. Childs and N. Balakrishnan, Conditional inference procedures for The Laplace distribution when the observed samples are progressively censored, Metrika 52 (3), 253–265, 2000.
• [12] U.J. Dixit and M.J. Nooghabi, Bayesian inference for the Pareto lifetime model in the presence of outliers under progressive censoring with binomial removals, Hacet. J. Math. Stat. 46 (5), 887–906, 2017.
• [13] D.D. Dyer and C.W. Whisenand, Best linear unbiased estimator of the parameter of the Rayleigh distribution-Part I: Small sample theory for censored order statistics, IEEE Transactions on Reliability 46 (1), 27–34, 1973.
• [14] F.E. Grubbs, Procedures for detecting outlying observations in samples, Technometrics 11 (1), 1–21, 1969.
• [15] D.M. Hawkins, Identification of outliers 11, Chapman and Hall, London, 1980.
• [16] B.K. Kale, Outliers-A review, J. Indian Statist. Assoc. 17, 51–67, 1979.
• [17] V.B. Kale and B.K. Kale, Outliers in exponential sample−A Bayesian approach, Gujarat Statistical Review, 1992.
• [18] A. Kohansal, Large Estimation of the stress-strength reliability of progressively censored inverted exponentiated Rayleigh distributions, J. Appl. Math. Stat. Inform. 13 (1), 49–76, 2017.
• [19] N.R. Mann, Best linear invariant estimation for Weibull parameters under progressive censoring, Technometrics 13 (3), 521–533, 1971.
• [20] M. Saleem and M. Aslam, On prior selection for the mixture of Rayleigh distribution using predictive Intervals, Pakistan J. Statist. 24 (1), 21–35, 2008.
• [21] M. Saleem and M. Aslam, On Bayesian analysis of the Rayleigh survival time assuming the random censor time, Pakistan J. Statist. 25 (2), 71–82, 2009.
• [22] A.M. Sarhan and A. Abuammoh, Statistical inference using progressively type-II censored data with random scheme, Int. Math. Forum (Online) 35 (3), 1713–1725, 2008.
• [23] R. Shanker, F. Hagos and S. Sujatha, On modeling of Lifetimes data using exponential and Lindley distributions, Biom Biostat Int J 2 (5), 1-9, 2015.
• [24] A.A. Soliman, Estimation of parameters of life from progressively censored data using Burr-XII model, IEEE Transactions on Reliability 54 (1), 34–42, 2005.
• [25] S.J. Wu and C.T. Chang, Parameter estimations based on exponential progressive type II censored data with binomial removals, Int. J. Inf. Manag. Sci. 13 (3), 37–46, 2002.
• [26] H.K. Yuen and S.K. Tse, Parameters estimation for Weibull distributed lifetimes under progressive censoring with random removals, J. Stat. Comput. Simul. 55 (1-2), 57–71, 1996.