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Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.


Bayes estimator Exponentiated Chen distribution Maximum likelihood estimator Metropolis-Hastings sampler Optimum censoring scheme Type-II progressive censoring

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Elshahhat, A., & Rastogi, M. K. (2022). Bayesian Life Analysis of Generalized Chen’s Population Under Progressive Censoring: Generalized Chen’s Population Under Progressive Censoring. Pakistan Journal of Statistics and Operation Research, 18(3), 675-702.


  1. Ahmed, E. A. (2014). Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach. Journal of Applied Statistics, 41(4), 752–768.
  2. Ashour, S. K., El-Sheikh, A. A., & Elshahhat, A. (2020). Inferences and optimal censoring Schemes for progressively first-failure censored Nadarajah-Haghighi distribution. Sankhya A, 1–39,
  3. Balakrishnan, N., & Cramer, E. (2014). The Art of Progressive Censoring. New York, USA: Springer, Birkhäuser.
  4. Balakrishnan, N., & Sandhu, R. A. (1995). A simple simulational algorithm for generating progressive Type-II censored samples. The American Statistician, 49(2), 229–230.
  5. Brown, L. (1968). Inadmissibility of the usual estimators of scale parameters in problems with unknown location and scale parameters. The Annals of Mathematical Statistics, 39(1), 29–48.
  6. Chaubey, Y. P., & Zhang, R. (2015). An extension of Chen’s family of survival distributions with bathtub shape or increasing hazard rate function. Communications in Statistics-Theory and Methods, 44(19), 4049–4064.
  7. Chen, M. H., & Shao, Q. M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1), 69–92.
  8. Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics & Probability Letters, 49(2), 155–161.
  9. Cohen, A. C. (1965). Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples. Technometrics, 7(4), 579–588.
  10. Dey, S., Kumar, D., Ramos, P. L., & Louzada, F. (2017). Exponentiated Chen distribution: properties and estimation. Communications in Statistics-Simulation and Computation, 46(10), 8118–8139.
  11. Dube, M., Krishna, H., & Garg, R. (2016). Generalized inverted exponential distribution under progressive first-failure censoring. Journal of Statistical Computation and Simulation, 86(6), 1095–1114.
  12. Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve. Journal of the American statistical Association, 83(402), 414–425.
  13. Elshahhat, A., & Abu El Azm, W. S. (2022). Statistical reliability analysis of electronic devices using generalized progressively hybrid censoring plan. Quality and Reliability Engineering International, 38, 1112–1130.
  14. Elshahhat, A., & Nassar, M. (2021). Bayesian survival analysis for adaptive Type-II progressive hybrid censored Hjorth data. Computational Statistics, 36(3), 1965–1990.
  15. Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. In a letter to Francis Baily, Esq. FRS &c. Philosophical transactions of the Royal Society of London, (115), 513–583.
  16. Greene, W. H. (2012). Econometric Analysis. New Jersey, USA: Pearson Prentice-Hall, Upper Saddle River.
  17. Gupta, R. C., Gupta, P. L., & Gupta, R. D. (1998). Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods, 27(4), 887–904.
  18. Gupta, R. D., & Kundu, D. (1999). Theory and methods: generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), 173–188.
  19. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications.
  20. Henningsen, A., & Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics, 26(3), 443–458.
  21. Hjorth, U. (1980). A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates. Technometrics, 22(1), 99–107.
  22. Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous Univariate Distributions (τ. 289). John wiley & sons.
  23. Kang, S. B., & Seo, J. I. (2011). Estimation in an exponentiated half logistic distribution under progressively Type-II censoring. Communications for Statistical Applications and Methods, 18(5), 657–666.
  24. Krishnamoorthy, K., & Lin, Y. (2010). Confidence limits for stress-strength reliability involving Weibull models. Journal of Statistical Planning and Inference, 140(7), 1754–1764.
  25. Kundu, D. (2008). Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics, 50(2), 144–154.
  26. Lawless, J. F. (2003). Statistical models and methods for lifetime data (τ. 362). John Wiley & Sons.
  27. Lee, K., & Cho, Y. (2017). Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring. Journal of Applied Statistics, 44(5), 811–832.
  28. Marinho, P. R. D., Silva, R. B., Bourguignon, M., Cordeiro, G. M., & Nadarajah, S. (2019). AdequacyModel: An R package for probability distributions and general purpose optimization. PloS one, 14(8), e0221487.
  29. Martz, H. F., & Waller, R. (1982). Bayesian Reliability Analysis. JOHN WILEY & SONS, INC. , 605 THIRD AVE. , NEW YORK, NY 10158, 1982, 704.
  30. Meeker, W. Q., & Escobar, L. A. (2014). Statistical Methods for Reliability Data. John Wiley & Sons.
  31. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. The journal of chemical physics, 21(6), 1087–1092.
  32. Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6), 543–558.
  33. Ng, H. K. T., Chan, P. S., & Balakrishnan, N. (2004). Optimal progressive censoring plans for the Weibull distribution. Technometrics, 46(4), 470–481.
  34. Pickands, J., III. (1975). Statistical inference using extreme order statistics. Annals of statistics, 3(1), 119–131.
  35. Plummer, M., Best, N., Cowles, K., & Vines, K. (2006). CODA: convergence diagnosis and output analysis for MCMC. R news, 6(1), 7–11.
  36. Pradhan, B., & Kundu, D. (2009). On progressively censored generalized exponential distribution. Test, 18(3), 497–515.
  37. Pradhan, B., & Kundu, D. (2013). Inference and optimal censoring schemes for progressively censored Birnbaum--Saunders distribution. Journal of Statistical Planning and Inference, 143(6), 1098–1108.
  38. Rastogi, M. K., Tripathi, Y. M., & Wu, S. J. (2012). Estimating the parameters of a bathtub-shaped distribution under progressive Type-II censoring. Journal of Applied Statistics, 39(11), 2389–2411.
  39. Rodrigues, J., & Zellner, A. (1994). Weighted balanced loss function and estimation of the mean time to failure. Communications in Statistics-Theory and Methods, 23(12), 3609–3616.
  40. Sarhan, A. M., & Apaloo, J. (2015). Inferences for a two-parameter lifetime distribution with bathtub shaped hazard based on censored data. International Journal of Statistics and Probability, 4(4), 77–92.
  41. Sarhan, A. M., Hamilton, D. C., & Smith, B. (2012). Parameter estimation for a two-parameter bathtub-shaped lifetime distribution. Applied Mathematical Modelling, 36(11), 5380–5392.
  42. Sen, T., Tripathi, Y. M., & Bhattacharya, R. (2018). Statistical inference and optimum life testing plans under Type-II hybrid censoring scheme. Annals of Data Science, 5(4), 679–708.
  43. Seo, J. I., Kang, S. B., & Kim, Y. (2017). Robust Bayesian estimation of a bathtub-shaped distribution under progressive Type-II censoring. Communications in Statistics-Simulation and Computation, 46(2), 1008–1023.
  44. Sharma, V. K. (2018). Bayesian analysis of head and neck cancer data using generalized inverse Lindley stress--strength reliability model. Communications in Statistics-Theory and Methods, 47(5), 1155–1180.
  45. Sharma, V. K., Singh, S. K., Singh, U., & Agiwal, V. (2015). The inverse Lindley distribution: a stress-strength reliability model with application to head and neck cancer data. Journal of Industrial and Production Engineering, 32(3), 162–173.
  46. Sultan, K. S., Alsadat, N. H., & Kundu, D. (2014). Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive Type-II censoring. Journal of Statistical Computation and Simulation, 84(10), 2248–2265.
  47. Surles, J. G., & Padgett, W. J. (2001). Inference for reliability and stress-strength for a scaled Burr Type X distribution. Lifetime data analysis, 7(2), 187–200.
  48. Tummala, V. M. R., & Sathe, P. T. (1978). Minimum expected loss estimators of reliability and parameters of certain lifetime distributions. IEEE Transactions on Reliability, 27(4), 283–285.
  49. Vishwakarma, P. K., Kaushik, A., Pandey, A., Singh, U., & Singh, S. K. (2018). Bayesian estimation for inverse Weibull distribution under progressive Type-II censored data with beta-binomial removals. Austrian Journal of Statistics, 47(1), 77–94.
  50. Weibull, W. (1951). A statistical distribution function of wide applicability. Journal of applied mechanics, 18(3), 293–297.
  51. Wu, S. J. (2008). Estimation of the two-parameter bathtub-shaped lifetime distribution with progressive censoring. Journal of Applied Statistics, 35(10), 1139–1150.
  52. Xie, M., Tang, Y., & Goh, T. N. (2002). A modified Weibull extension with bathtub-shaped failure rate function. Reliability Engineering and System Safety, 76(3), 279–285.