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Bayesian analysis for lognormal distribution under progressive Type-II censoring

Year 2019, Volume: 48 Issue: 5, 1488 - 1504, 08.10.2019

Abstract

In this paper, we consider the problems of Bayesian estimation and prediction for lognormal distribution under progressive Type-II censored data. We propose various non-informative and informative priors for the unknown lognormal parameters and compute the Bayes estimates under squared error loss function. Importance sampling technique and OpenBUGS are taken into consideration for the computational purpose. Further, we predict lifetimes of both censored and future samples under one- and two-sample prediction frameworks. We also compute the corresponding Bayes predictive bounds. A simulation study is conducted to compare the performance of proposed estimates and a real data set is analyzed to illustrate applications of this study. Finally, a conclusion is presented.

References

  • [1] E.K. Al-Hussaini, Predicting observables from a general class of distributions, J. Statist. Plann. Inference 79, 79-91, 1999.
  • [2] M.A.M. Ali Mousa, Inference and prediction for Pareto progressively censored data, J. Stat. Comput. Simul. 71, 163-181, 2001.
  • [3] M.A.M. Ali Mousa and S.A. Al-Sagheer, Bayesian prediction for progressively type-II censored data from the Rayleigh model, Comm. Statist. Theory Methods 34, 2353- 2361, 2005.
  • [4] M.A.M. Ali Mousa and Z.F. Jaheen, Bayesian prediction for progressively censored data from the Burr model, Statist. Papers 43, 587-593, 2002.
  • [5] A. Asgharzadeh, R. Valiollahi and D. Kundu, Prediction for future failures in Weibull distribution under hybrid censoring, J. Stat. Comput. Simul. 85, 824-838, 2015.
  • [6] N. Balakrishnan and E. Cramer, The Art of Progressive Censoring: Applications to Reliability and Quality, Birkhäuser, 2014.
  • [7] N. Balakrishnan, N. Kannan, C.T. Lin and H.K.T. Ng, Point and interval estima- tion for Gaussian distribution, based on progressively type-II censored samples, IEEE Trans. Reliab. 52, 90-95, 2003.
  • [8] N. Balakrishnan and J. Mi, Existence and uniqueness of the MLEs for normal distri- bution based on general progressively type-II censored samples, Statist. Probab. Lett. 64, 407-414, 2003.
  • [9] A. Banerjee and D. Kundu, Inference based on type-II hybrid censored data from a Weibull distribution, IEEE Trans. Reliab. 57, 369-378, 2008.
  • [10] A. Basavalingappa, J.M. Passage, M.Y. Shen and J.R. Lloyd, Lognormal versus Weibull distribution, In IEEE International Integrated Reliability Workshop (IIRW), 2017.
  • [11] M.-H. Chen and Q.-M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist. 8, 69-92, 1999.
  • [12] E.L. Crow and L. Shimizu, Lognormal Distributions: Theory and Applications, Dekker, 1988.
  • [13] S. Dey, S. Singh, Y.M. Tripathi and A. Asgharzadeh, Estimation and prediction for a progressively censored generalized inverted exponential distribution, Stat. Methodol. 32, 185-202, 2016.
  • [14] S.-R. Huang and S.-J. Wu, Bayesian estimation and prediction for Weibull model with progressive censoring, J. Stat. Comput. Simul. 82, 1607-1620, 2012.
  • [15] R.J. Hyndman, Computing and graphing highest density regions, Amer. Statist. 50, 120-126, 1996.
  • [16] Z.F. Jaheen, Prediction of progressive censored data from the Gompertz model, Comm. Statist. Simulation Comput. 32, 663-676, 2003.
  • [17] X. Jia, S. Nadarajah and B. Guo, The effect of mis-specification on mean and selection between the Weibull and lognormal models, Phys. A 492, 1875-1891, 2018.
  • [18] N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, second edition, Wiley, 1994.
  • [19] D. Kelly and C. Smith, Bayesian Inference for Probabilistic Risk Assessment: A Practitioner’s Guidebook, Springer, 2011.
  • [20] H.M. Khan, M.S. Haq and S.B. Provost, Bayesian prediction for the log-normal model under type II censoring, J. Stat. Theory Appl. 5, 141-160, 2006.
  • [21] D. Kundu, Bayesian inference and life testing plan for Weibull distribution in presence of progressive censoring, Technometrics 50, 144-154, 2008.
  • [22] D. Kundu and H. Howlader, Bayesian inference and prediction of the inverse Weibull distribution for type-II censored data, Comput. Statist. Data Anal. 54, 1547-1558, 2010.
  • [23] D. Kundu and M.Z. Raqab, Bayesian inference and prediction of order statistics for a type-II censored Weibull distribution, J. Statist. Plann. Inference 142, 41-47, 2012.
  • [24] J. Lawless, Statistical Models and Methods for Lifetime Data, second edition, Wiley, 2003.
  • [25] R.B. Leipnik, Lognormal random variables, J. Amer. Math. Soc., Series B 32, 327-347, 1991.
  • [26] D.V. Lindley, Approximate Bayesian methods, Trabajos de Estadistica 31, 223-245, 1980.
  • [27] W.Q. Meeker and L.A. Escobar, Statistical Methods and Reliability Data, Wiley, 1998.
  • [28] M.M. Mohie El-Din and A.R. Shafay, One- and two-sample Bayesian prediction in- tervals based on progressively type-II censored data, Statist. Papers 54, 287-307, 2013.
  • [29] H.K.T. Ng, P.S. Chan and N. Balakrishnan, Estimation of parameters from progres- sively censored data using EM algorithm, Comput. Statist. Data Anal. 39, 371-386, 2002.
  • [30] H.K.T. Ng, P.S. Chan and N. Balakrishnan, Optimal progressive censoring plans for the Weibull distribution, Technometrics 46, 470-481, 2004.
  • [31] H. Panahi and A. Sayyareh, Parameter estimation and prediction of order statistics for the Burr Type XII distribution with type II censoring, J. Appl. Stat. 41, 215-232, 2014.
  • [32] B. Pradhan and D. Kundu, On progressively censored generalized exponential distri- bution, TEST 18, 497-515, 2009.
  • [33] M.Z. Raqab, S.A. Al-Awadhi and D. Kundu, Discriminating among Weibull, log- normal, and log-logistic distributions, Comm. Statist. Simulation Comput. 47, 1397- 1419, 2018.
  • [34] S. Singh and Y.M. Tripathi, Bayesian estimation and prediction for a hybrid censored lognormal distribution, IEEE Trans. Reliab. 65, 782-795, 2016.
  • [35] S. Singh and Y.M. Tripathi, Estimating the parameters of an inverse Weibull distri- bution under progressive type-I interval censoring, Statist. Papers 59, 21-56, 2018.
  • [36] S. Singh, Y.M. Tripathi and S.-J. Wu, On estimating parameters of a progressively censored lognormal distribution, J. Stat. Comput. Simul. 85, 1071-1089, 2015.
  • [37] S.K. Sinha, Bayesian Estimation, New Age International Publishers, 1998.
  • [38] A.A. Soliman, Estimation of parameters of life from progressively censored data using Burr-XII model, IEEE Trans. Reliab. 54, 34-42, 2005.
  • [39] L. Tierney and J.B. Kadane, Accurate approximations for posterior moments and marginal densities, J. Amer. Statist. Assoc. 81, 82-86, 1986.
  • [40] N. Turkkan and T. Pham-Gia, Computation of the highest posterior density interval in Bayesian analysis, J. Stat. Comput. Simul. 44, 243-250, 1993.
Year 2019, Volume: 48 Issue: 5, 1488 - 1504, 08.10.2019

Abstract

References

  • [1] E.K. Al-Hussaini, Predicting observables from a general class of distributions, J. Statist. Plann. Inference 79, 79-91, 1999.
  • [2] M.A.M. Ali Mousa, Inference and prediction for Pareto progressively censored data, J. Stat. Comput. Simul. 71, 163-181, 2001.
  • [3] M.A.M. Ali Mousa and S.A. Al-Sagheer, Bayesian prediction for progressively type-II censored data from the Rayleigh model, Comm. Statist. Theory Methods 34, 2353- 2361, 2005.
  • [4] M.A.M. Ali Mousa and Z.F. Jaheen, Bayesian prediction for progressively censored data from the Burr model, Statist. Papers 43, 587-593, 2002.
  • [5] A. Asgharzadeh, R. Valiollahi and D. Kundu, Prediction for future failures in Weibull distribution under hybrid censoring, J. Stat. Comput. Simul. 85, 824-838, 2015.
  • [6] N. Balakrishnan and E. Cramer, The Art of Progressive Censoring: Applications to Reliability and Quality, Birkhäuser, 2014.
  • [7] N. Balakrishnan, N. Kannan, C.T. Lin and H.K.T. Ng, Point and interval estima- tion for Gaussian distribution, based on progressively type-II censored samples, IEEE Trans. Reliab. 52, 90-95, 2003.
  • [8] N. Balakrishnan and J. Mi, Existence and uniqueness of the MLEs for normal distri- bution based on general progressively type-II censored samples, Statist. Probab. Lett. 64, 407-414, 2003.
  • [9] A. Banerjee and D. Kundu, Inference based on type-II hybrid censored data from a Weibull distribution, IEEE Trans. Reliab. 57, 369-378, 2008.
  • [10] A. Basavalingappa, J.M. Passage, M.Y. Shen and J.R. Lloyd, Lognormal versus Weibull distribution, In IEEE International Integrated Reliability Workshop (IIRW), 2017.
  • [11] M.-H. Chen and Q.-M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist. 8, 69-92, 1999.
  • [12] E.L. Crow and L. Shimizu, Lognormal Distributions: Theory and Applications, Dekker, 1988.
  • [13] S. Dey, S. Singh, Y.M. Tripathi and A. Asgharzadeh, Estimation and prediction for a progressively censored generalized inverted exponential distribution, Stat. Methodol. 32, 185-202, 2016.
  • [14] S.-R. Huang and S.-J. Wu, Bayesian estimation and prediction for Weibull model with progressive censoring, J. Stat. Comput. Simul. 82, 1607-1620, 2012.
  • [15] R.J. Hyndman, Computing and graphing highest density regions, Amer. Statist. 50, 120-126, 1996.
  • [16] Z.F. Jaheen, Prediction of progressive censored data from the Gompertz model, Comm. Statist. Simulation Comput. 32, 663-676, 2003.
  • [17] X. Jia, S. Nadarajah and B. Guo, The effect of mis-specification on mean and selection between the Weibull and lognormal models, Phys. A 492, 1875-1891, 2018.
  • [18] N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, second edition, Wiley, 1994.
  • [19] D. Kelly and C. Smith, Bayesian Inference for Probabilistic Risk Assessment: A Practitioner’s Guidebook, Springer, 2011.
  • [20] H.M. Khan, M.S. Haq and S.B. Provost, Bayesian prediction for the log-normal model under type II censoring, J. Stat. Theory Appl. 5, 141-160, 2006.
  • [21] D. Kundu, Bayesian inference and life testing plan for Weibull distribution in presence of progressive censoring, Technometrics 50, 144-154, 2008.
  • [22] D. Kundu and H. Howlader, Bayesian inference and prediction of the inverse Weibull distribution for type-II censored data, Comput. Statist. Data Anal. 54, 1547-1558, 2010.
  • [23] D. Kundu and M.Z. Raqab, Bayesian inference and prediction of order statistics for a type-II censored Weibull distribution, J. Statist. Plann. Inference 142, 41-47, 2012.
  • [24] J. Lawless, Statistical Models and Methods for Lifetime Data, second edition, Wiley, 2003.
  • [25] R.B. Leipnik, Lognormal random variables, J. Amer. Math. Soc., Series B 32, 327-347, 1991.
  • [26] D.V. Lindley, Approximate Bayesian methods, Trabajos de Estadistica 31, 223-245, 1980.
  • [27] W.Q. Meeker and L.A. Escobar, Statistical Methods and Reliability Data, Wiley, 1998.
  • [28] M.M. Mohie El-Din and A.R. Shafay, One- and two-sample Bayesian prediction in- tervals based on progressively type-II censored data, Statist. Papers 54, 287-307, 2013.
  • [29] H.K.T. Ng, P.S. Chan and N. Balakrishnan, Estimation of parameters from progres- sively censored data using EM algorithm, Comput. Statist. Data Anal. 39, 371-386, 2002.
  • [30] H.K.T. Ng, P.S. Chan and N. Balakrishnan, Optimal progressive censoring plans for the Weibull distribution, Technometrics 46, 470-481, 2004.
  • [31] H. Panahi and A. Sayyareh, Parameter estimation and prediction of order statistics for the Burr Type XII distribution with type II censoring, J. Appl. Stat. 41, 215-232, 2014.
  • [32] B. Pradhan and D. Kundu, On progressively censored generalized exponential distri- bution, TEST 18, 497-515, 2009.
  • [33] M.Z. Raqab, S.A. Al-Awadhi and D. Kundu, Discriminating among Weibull, log- normal, and log-logistic distributions, Comm. Statist. Simulation Comput. 47, 1397- 1419, 2018.
  • [34] S. Singh and Y.M. Tripathi, Bayesian estimation and prediction for a hybrid censored lognormal distribution, IEEE Trans. Reliab. 65, 782-795, 2016.
  • [35] S. Singh and Y.M. Tripathi, Estimating the parameters of an inverse Weibull distri- bution under progressive type-I interval censoring, Statist. Papers 59, 21-56, 2018.
  • [36] S. Singh, Y.M. Tripathi and S.-J. Wu, On estimating parameters of a progressively censored lognormal distribution, J. Stat. Comput. Simul. 85, 1071-1089, 2015.
  • [37] S.K. Sinha, Bayesian Estimation, New Age International Publishers, 1998.
  • [38] A.A. Soliman, Estimation of parameters of life from progressively censored data using Burr-XII model, IEEE Trans. Reliab. 54, 34-42, 2005.
  • [39] L. Tierney and J.B. Kadane, Accurate approximations for posterior moments and marginal densities, J. Amer. Statist. Assoc. 81, 82-86, 1986.
  • [40] N. Turkkan and T. Pham-Gia, Computation of the highest posterior density interval in Bayesian analysis, J. Stat. Comput. Simul. 44, 243-250, 1993.
There are 40 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Sukhdev Singh This is me 0000-0001-6282-4281

Yogesh Mani Tripathi 0000-0002-9687-6036

Shuo-jye Wu 0000-0001-7294-8018

Publication Date October 8, 2019
Published in Issue Year 2019 Volume: 48 Issue: 5

Cite

APA Singh, S., Tripathi, Y. M., & Wu, S.-j. (2019). Bayesian analysis for lognormal distribution under progressive Type-II censoring. Hacettepe Journal of Mathematics and Statistics, 48(5), 1488-1504.
AMA Singh S, Tripathi YM, Wu Sj. Bayesian analysis for lognormal distribution under progressive Type-II censoring. Hacettepe Journal of Mathematics and Statistics. October 2019;48(5):1488-1504.
Chicago Singh, Sukhdev, Yogesh Mani Tripathi, and Shuo-jye Wu. “Bayesian Analysis for Lognormal Distribution under Progressive Type-II Censoring”. Hacettepe Journal of Mathematics and Statistics 48, no. 5 (October 2019): 1488-1504.
EndNote Singh S, Tripathi YM, Wu S-j (October 1, 2019) Bayesian analysis for lognormal distribution under progressive Type-II censoring. Hacettepe Journal of Mathematics and Statistics 48 5 1488–1504.
IEEE S. Singh, Y. M. Tripathi, and S.-j. Wu, “Bayesian analysis for lognormal distribution under progressive Type-II censoring”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, pp. 1488–1504, 2019.
ISNAD Singh, Sukhdev et al. “Bayesian Analysis for Lognormal Distribution under Progressive Type-II Censoring”. Hacettepe Journal of Mathematics and Statistics 48/5 (October 2019), 1488-1504.
JAMA Singh S, Tripathi YM, Wu S-j. Bayesian analysis for lognormal distribution under progressive Type-II censoring. Hacettepe Journal of Mathematics and Statistics. 2019;48:1488–1504.
MLA Singh, Sukhdev et al. “Bayesian Analysis for Lognormal Distribution under Progressive Type-II Censoring”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 5, 2019, pp. 1488-04.
Vancouver Singh S, Tripathi YM, Wu S-j. Bayesian analysis for lognormal distribution under progressive Type-II censoring. Hacettepe Journal of Mathematics and Statistics. 2019;48(5):1488-504.