Estimation and prediction for the XLindley distribution under progressively Type-II censoring

Document Type : Original Scientific Paper

Authors

Department of Statistics, Payame Noor University, Tehran, Iran

10.22034/jsmta.2026.24546.1217

Abstract

In this paper‎, ‎we consider the problems of estimating unknown parameters as well as predicting the failure times of the removed units in multiple stages of the progressively censored sample coming from an XLindley distribution‎. ‎We obtain maximum likelihood‎, ‎moment-based‎, ‎bootstrap‎, ‎and Bayes estimates under the squared error loss function‎. ‎Since the integrals associated with the Bayesian estimators do not admit closed-form solutions‎, ‎Lindley's approximation‎, ‎the Markov chain Monte Carlo method‎, ‎and Tierney and Kadane's approximation are used to approximate these integrals‎. ‎We also compute the confidence intervals based on the asymptotic distribution of the maximum likelihood estimate‎, ‎the confidence intervals based on the asymptotic distribution of the logarithm of the maximum likelihood estimate‎, ‎the exact confidence intervals‎, ‎the confidence intervals based on the likelihood ratio statistic‎, ‎the bootstrap confidence intervals‎, ‎and the Bayesian confidence intervals‎. ‎Different point and interval predictors are derived based on classical and Bayesian approaches‎. ‎A real example is provided to further explain the methods introduced‎. ‎A Monte Carlo simulation study is conducted to evaluate and compare the performance of different estimation and prediction methods.

Keywords

Main Subjects

References

Alotaibi, R., Nassar, M. and Elshahhat, A. (2022). Computational analysis of XLindley parameters using adaptive Type-II progressive hybrid censoring with applications in chemical engineering. Mathematics, 10(18):3325.
Alotaibi, R., Nassar, M. and Elshahhat, A. (2023). Reliability estimation under normal operating conditions for progressively type-II XLindley censored data. Axioms, 12(4):352.
Alotaibi, R., Nassar, M. and Elshahhat, A. (2024). Data analysis with applications in physics and engineering using XLindley model with improved adaptive Type-II progressively censored samples. Heliyon, 10(14):e33598.
Balakrishnan, N. and Aggarwala, R. (2000). Progressive Censoring: Theory Methods and Applications. Boston: Birkhäuser.
Balakrishnan, N. and Cramer, E. (2014). Art of Progressive Censoring: Applications to Reliability and Quality. New York: Birkhauser.
Casella, G. and Berger, R.L. (2002). Statistical Inference. 2nd ed., CRC Press.
Chen, M.H. and Shao, Q.M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1):69–92.
Chivers, C. (2012). MHadaptive: General Markov Chain Monte Carlo for Bayesian Inference using adaptive Metropolis-Hastings sampling. R package.
Chouia, S. and Zeghdoudi, H. (2021). The XLindley distribution: properties and application. Journal of Statistical Theory and Applications, 20(2):318–327.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics, 7(1):1–26.
Elbatal, I., Elshahhat, A., Semary, H.E. and Nassar, M. (2025). Reliability assessment of two production lines using joint progressively type-II censored XLindley samples. Scientific Reports, 15(1):26841.
Elshahhat, A. and Alotaibi, R. (2026). The XLindley survival model under generalized progressively censored data: theory, inference, and applications. Axioms, 15(1):56.
Gelman, A. and Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7(4):457–472.
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to the calculations of posterior moments. Bayesian Statistics, 4:641–649.
Hasselman, B. (2018). nleqslv: solve systems of nonlinear equations, R package version 3.3.2.
Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57:97–109.
Heidelberger, P. and Welch, P.D. (1983). Simulation run length control in the presence of an initial transient. Operations Research, 31(6):1109–1144.
Kaminsky, K.S. and Rhodin, L.S. (1985). Maximum likelihood prediction. Annals of the Institute of Statistical Mathematics, 37(Part A):507–517.
Lindley, D.V. (1980). Approximate Bayesian methods. Trabajos de estadística y de investigación operativa, 31(1):223–237.
Meeker, W.Q. and Escobar, L.A. (1998). Statistical Methods for Reliability Data. New York: Wiley.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6):1087–1092.
Nassar, M., Alotaibi, R. and Elshahhat, A. (2023). Reliability estimation of XLindley constant-stress partially accelerated life tests using progressively censored samples. Mathematics, 11(6):1331.
Raftery, A.E. and Lewis, S.M. (1992). Comment: One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo. Statistical Science, 7(4):493–497.
Raftery, A.E. and Lewis, S.M. (1996). Implementing MCMC. In Markov Chain Monte Carlo in Practice, Gilks, W.R., Richardson, S. and Spiegelhalter, D.J., pp. 115–130, Boca Raton: Chapman and Hall/CRC.
Raqab, M.Z. and Nagaraja, H.N. (1995). On some predictors of future order statistics. Metron, 53(12):185–204.
R Core Team (2026). R: A Language and Environment for Statistical Computing. Austria, Vienna: R Foundation for Statistical Computing. https://www.R-project.org/
Robert, C. and Casella, G. (2004). Monte Carlo Statistical Methods. 2nd edition, New York: Springer.
Soetaert, K. (2009). rootSolve: Nonlinear root finding, equilibrium and steady-state analysis of ordinary differential equations. R-package version 1.8.2.
Stablein, D.M., Carter, J. and Novak, J.W. (1981). Analysis of survival data with nonproportional hazard functions. Controlled Clinical Trials, 2(2):149–159.
Thongjub, N. and Panichkitkosolkul, W. (2026). Improved inference for the XLindley distribution: A comparative analysis of confidence intervals and applications. Statistics Optimization and Information Computing, 15(6):4900–4921.
Tierney, L. and Kadane, J.B. (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81(393):82–86.
Zanjiran, F. and MirMostafaee, S.M.T.K. (2024). On estimation and prediction for the XLindley distribution based on record data. Reliability: Theory and Applications, 19(2 (78)):258–272.
Journal of Statistical Modelling: Theory and Applications
Volume 7, Issue 1
January 2026
Pages 55-85

Files

History

  • Receive Date: 09 May 2026
  • Revise Date: 24 August 2026
  • Accept Date: 23 September 2026

Share

How to cite

Statistics