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This article proposes the Bayes estimation of the parameter and reliability function for xgamma distribution in the presence of type-I hybrid censored observations. The Bayes estimate of the parameter has been obtained by assuming informative and non-informative priors using general entropy loss function. Obviously, censoring adds difficulties in estimation procedure; hence the Bayes estimators computed with type-I hybrid censored observation under the mentioned prior often do not assume any standard form. Therefore, Bayes estimates are computed using Tierney-Kadane approximation and Markov Chain Monte Carlo numerical technique. Further, different interval estimates namely asymptotic confidence interval, bootstrap confidence interval and highest posterior density interval along with the width of the interval and coverage probability are also discussed. The maximum likelihood estimate for the same has also been computed using non- linear maximization iterative procedure and compared with corresponding Bayes estimates using Monte Carlo simulations. The comparison of the estimators are made in terms of average loss over whole sample space and corresponding length of the interval. lastly, one medical data set has been considered for the real application of the proposed study.


xgamma distribution type-I hybrid censoring point and interval estimation Bayes computation

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How to Cite
Yadav, A. S. (2023). Bayesian estimation for the type-I hybrid xgamma distribution using asymmetric loss function. Pakistan Journal of Statistics and Operation Research, 19(1), 27-49.


  1. Balakrishnan, N. and Kundu, D. (2013). Hybrid censoring: models, inferential results and applications.
  2. Computational Statistics and Data Analysis, 57(1):166–209.
  3. Basu, A. P. and Ebrahimi, N. (1991). Bayesian approach to life testing and reliability estimation using asymmetric loss function. J. Statist. Plann. Infer., 29:21–31.
  4. Bjerkedal, T. (1960). Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Hygiene, 72(1):130–48.
  5. Chadli, A. and Kermoune, S. (2021). Reliability estimation in a rayleigh pareto model with progressively type-ii right censored data. Pakistan Journal of Statistics and Operation Research, 17(3):729–743, doi:
  6. Chen, M. H. and Shao, Q. M. (1999). Monte carlo estimation of bayesian credible and hpd intervals. Journal of Computational and Graphical Statistics.
  7. Dey, S. and Pradhan, B. (2014). Generalized inverted exponential distribution under hybrid censoring. Statistical methodology, 18:101–114.
  8. Draper, N. and Guttman, I. (1987). Bayesian analysis of hybrid life tests with exponential failure times. Ann. Inst. Statist. Math., 39:219–225.
  9. Ebrahimi, N. (1990). Estimating the parameter of an exponential distribution from hybrid life test. J. Statist. Plann. Inference., 23:255–261.
  10. Efron, B. and Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, 1(1):54–75.
  11. Epstein, B. (1954). Truncated life test in exponential case. Ann. Math. Statistics, 25:555–564.
  12. Geman, S. and Geman, A. (1984). Stochastic relaxation, gibbs distributions and the bayesian restoration of images. IEEE Trans. Pattern Analysis and Machine Intelligence, 6:721–740.
  13. Gupta, R. and Kundu, D. (1998). Hybrid censoring schemes with exponential failure distribution. Commun. Statist. Theor. Meth., 27:3065–3083.
  14. Hastings, W. K. (1970). Monte carlo sampling methods using markov chains and their applications. Biometrika, 57(1):97–109.
  15. Kamal, R. M. and Ismail, M. A. (2021). Estimation for flexible weibull extension-burr xii distribution under adaptive type-ii progressive censoring scheme. Pakistan Journal of Statistics and Operation Research, 17(4):1065 1094. doi:
  16. Kundu, D. and Pradhan, B. (2009). Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring. Comm. Statist. Theor. Meth., 38:2030–2041.
  17. Sen, S., Maiti, S. S., and Chandra, N. (2016). The xgamma distribution: statistical properties and application. Journal of Modern Applied Statistical Methods, 15(1), Article 38:10.22237/jmasm/1462077420.
  18. Sen, S., Maiti, S. S., and Chandra, N. (2018). Survival estimation in xgamma distribution under progres- sively type-ii right censored scheme. Model Assisted Statistics and Applications, 13:107–121.
  19. Singh, S. K., Singh, U., and Yadav, A. S. (2014). Parameter estimation in marshall-olkin exponential distribution under type-i hybrid censoring scheme. J. Stat. Appl. Pro., 3(2):1–11.
  20. Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the gibbs sampler and related markov chain monte carlo methods. Journal of the Royal Statistical Society Series B (Methodological), 55(1):3– 23.
  21. Talhi, H. and Aiachi, H. (2021). On truncated zeghdoudi distribution : Posterior analysis under different loss functions for type ii censored data. Pakistan Journal of Statistics and Operation Research, 17(2):497–508, doi:
  22. Tierney, L. and Kadane, J. B. (1986). Accurate approximations for posterior moments and marginal densi ties. Journal of the American Statistical Association, 81(393):82–86.
  23. Upadhyay, S. K., Vasishta, N., and Smith, A. F. M. (2001). Bayes inference in life testing and reliability via markov chain monte carlo simulation. Sankhya A, 63:15–40.
  24. Yadav, A. S., Saha, M., Singh, S. K., and Singh, U. (2018). Bayesian estimation of the parameter and the reliability characteristics of xgamma distribution using type-ii hybrid censored data. Life Cycle Reliability and Safety Engineering, 8(1):1–10 doi:–018–0065–5.
  25. Yadav, A. S., Singh, S. K., and Singh, U. (2016). On hybrid censored inverse lomax distribution: Application to the survival data. Statistica, 76:185–203.
  26. Zellner, A. (1986). A bayesian estimation and prediction using asymmetric loss function. JASA, 81:446–451.