Bayesian inference and optimal plan for the family of inverted exponentiated distributions under doubly censored data

Abstract

In this paper, we consider inference upon unknown parameters of the family of inverted exponentiated distributions when it is known that data are doubly censored. Maximum likelihood and Bayes estimates under different loss functions are derived for estimating the parameters. We use Metropolis-Hastings algorithm to draw Markov chain Monte Carlo samples, which are used to compute the Bayes estimates and construct the Bayesian credible intervals. Further, we present point and interval predictions of the censored data using the Bayesian approach. The performance of proposed methods of estimation and prediction are investigated using simulation studies, and two illustrative examples are discussed in support of the suggested methods. Finally, we propose the optimal plans under double censoring scheme.

Keywords

• Bayes estimate
• Bayesian prediction
• double censoring
• inverted exponentiated exponential distribution
• maximum likelihood estimate

References

1. [1] A.M. Abouammoh and A.M. Alshingiti,Reliability estimation of generalized inverted exponential distribution, J. Stat. Comput. Simul. 79 (11), 1301-1315, 2009.
2. [2] N. Balakrishnan,On the maximum likelihood estimation of the location and scale parameters of exponential distribution based on multiply type II censored samples, J. Appl. Stat. 17 (1), 55-61, 1990.
3. [3] N. Balakrishnan, N. Kannan, C.T. Lin and H.K.T. Ng, Point and interval estimation for Gaussian distribution, based on progressively type-II censored samples, IEEE Trans. Reliab. 52 (1), 90-95, 2003.
4. [4] A.P. Basu and N. Ebrahimi,Bayesian approach to life testing and reliability estimation using asymmetric loss function, J. Stat. Plan. Inference.29 (1-2), 21-31, 1991.
5. [5] R. Bhattacharya, B. Pradhan, and A. Dewanji, Optimum life testing plans in presence of hybrid censoring: A cost function approach, Appl. Stoch. Models Bus. Ind. 30 (5), 519-528, 2014.
6. [6] T. Choi, A.K.H. Kim and S. Choi, Semiparametric least-squares regression with doubly-censored data, Comput. Stat. Data Anal. 164, 107306, 2021,
7. [7] S. Dey and T. Dey,On progressively censored generalized inverted exponential distribution, J. Appl. Stat. 41 (12), 2557-2576, 2014.
8. [8] 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.

9. [9] B. Efron B and D.V. Hinkley,Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information, Biometrika. 65 (3), 457-487, 1978.
10. [10] A.J. Fernández, Bayesian inference from type II doubly censored Rayleigh data, Stat. Probab. Lett. 48 (4), 393-399, 2000.
11. [11] A.J. Fernández, Weibull inference using trimmed samples and prior information, Stat. Pap. 50, 119-136, 2009.
12. [12] N. Feroze and M. Aslam, Comparison of improved class of priors for the analysis of the Burr type VII model under doubly censored samples, Hacet. J. Math. Stat. 50 (5), 1509-1533, 2021.
13. [13] M.E. Ghitany, V.K. Tuan and N. Balakrishnan, Likelihood estimation for a general class of inverse exponentiated distributions based on complete and progressively censored data, J. Stat. Comput. Simul. 84 (1), 96-106, 2014.
14. [14] R.D. Gupta and D. Kundu, On the comparison of Fisher information of the Weibull and GE distributions, J. Stat. Plann. Inference 136 (9), 3130-3144, 2006.
15. [15] S.R.K. Iyengar and R.K. Jain, Numerical Methods, New Age International, 2009.
16. [16] T. Kayal, Y.M. Tripathi, D. Kundu and M.K. Rastogi, Statistical inference of Chen distribution based on type I progressive hybrid censored samples, Stat. Optim. Inf. Comput. 10 (2), 627-642, 2022.
17. [17] T. Kayal, Y.M. Tripathi and M.K. Rastogi, Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring, Commun. Stat. Theory Methods 47 (7), 1615-1640, 2018.
18. [18] M.S. Kotb and M.Z. Raqab, Inference and prediction for modified Weibull distribution based on doubly censored samples, Math. Comput. Simul. 132, 195-207, 2017.
19. [19] H. Krishna and K. Kumar, Reliability estimation in generalized inverted exponential distribution with progressively type II censored sample, J. Stat. Comput. Simul. 83 (6), 1007-1019, 2013.
20. [20] D. Kundu, Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring, Technometrics. 50 (2), 144-154, 2008.
21. [21] D. Kundu and B. Pradhan, Bayesian analysis of progressively censored competing risks data, Sankhya B. 73, 276-296, 2011.
22. [22] E.T. Lee, J. Wang, Statistical Methods for Survival Data Analysis, Wiley, 2003.
23. [23] C. Lodhi, Y.M. Tripathi, and M.K. Rastogi, Estimating the parameters of a truncated normal distribution under progressive type II censoring, Commun. Stat. Simul. Comput. 50 (9), 2757-2781, 2021.
24. [24] B. Long, Estimation and prediction for the Rayleigh distribution based on double type-I hybrid censored data, Commun. Stat. Simul. Comput. 52 (8), 3553-3567, 2023.
25. [25] R.K. Maurya, Y.M. Tripathi, T. Sen, and M.K. Rastogi, Inference for an inverted exponentiated Pareto distribution under progressive censoring, J. Stat. Theory Pract. 13, 1-32, 2019.
26. [26] R.K. Maurya, Y.M. Tripathi, T. Sen and M.K. Rastogi, On progressively censored inverted exponentiated Rayleigh distribution, J. Stat. Comput. Simul. 89 (3), 492- 518, 2019.
27. [27] S. Mondal, R. Bhattacharya, B. Pradhan and D. Kundu, Bayesian optimal life-testing plan under the balanced two sample type-II progressive censoring scheme, Appl. Stochastic Models Bus. Ind. 36 (4), 628-640, 2020.
28. [28] B. Pareek, D. Kundu and S. Kumar, On progressively censored competing risks data for Weibull distributions, Comput. Stat. Data Anal. 53 (12), 4083-4094, 2009.
29. [29] A. Parsian, N.S. Farsipour and N. Nematollahi, On the minimaxity of Pitman type estimator under a LINEX loss function, Commun. Stat. Theory Methods. 22 (1), 97-113, 1992.
30. [30] P.G.M. Peer, J.A. Van Dijck, A.L.M. Verbeek, J.H.C.L. Hendriks and R. Holland, Age-dependent growth rate of primary breast cancer, Cancer. 71 (11), 3547-3551, 1993.
31. [31] B. Pradhan and D. Kundu, On progressively censored generalized exponential distribution, Test. 18, 497-515, 2009.
32. [32] M.K. Rastogi and Y.M. Tripathi, Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring, J. Appl. Stat. 41 (11), 2375-2405, 2014.
33. [33] C.P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, 1999.
34. [34] T. Sen, S. Singh and Y.M. Tripathi, Statistical inference for lognormal distribution with type-I progressive hybrid censored data, Am. J. Math. Manag. Sci. 38 (1), 70-95, 2019.
35. [35] A.R. Shafay, N. Balakrishnan and Y. Abdel-Aty, Bayesian inference based on a jointly type-II censored sample from two exponential populations, J. Stat. Comput. Simul. 84 (11), 2427-2440, 2014.
36. [36] S.P. Sheng, The Cox-Aalen model for doubly censored data, Commun. Stat. Theory Methods 51 (23), 8075-8092, 2021.
37. [37] S. Singh, Y.M. Tripathi and S.-J. Wu, Bayesian analysis for lognormal distribution under progressive type-II censoring, Hacet. J. Math. Stat. 48 (5), 1488-1504, 2019.
38. [38] S.K. Singh, U. Singh and D. Kumar, Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and noninformative priors, J. Stat. Comput. Simul. 83 (12), 2258-2269, 2013.
39. [39] L. Wang, K. Wu, and X. Zuo, Inference and prediction of progressive Type-II censored data from unit-generalized Rayleigh distribution, Hacet. J. Math. Stat. 51 (6), 17521767, 2022,
40. [40] S.-J. Wu, S.R. Huang and J.H. Wang, Determination of warranty length for one-shot devices with Rayleigh lifetime distribution, Commun. Stat. Theory Methods 52 (5), 1400-1416, 2023.
41. [41] H.R. Varian, A Bayesian approach to real estate assessment. Studies in Bayesian econometrics and statistics in Honor of Leonard J. Savage, pages 195-208, 1975.