‎Expected Experimentation Time, Estimation and Prediction for the Power Lindley Distribution Based on Progressively Type II Censored Data ‎W‎ith Binomial Removals

Document Type : Original Paper


Department of Statistics, University of Mazandaran, Babolsar, Iran


In this paper, the problem of estimation and prediction for the power Lindley distribution is studied based on progressively Type-II censored data with binomial removals. First, we work on the estimation of the parameters of the power Lindley distribution with the help of maximum likelihood and Bayesian methods. The Bayesian estimation is done based on the symmetric squared error loss function and the asymmetric general entropy loss function. Since the Bayesian estimates involve integrals that do not seem to have explicit forms, we use the Metropolis-Hastings algorithm to approximate these integrals. A simulation study is presented to evaluate the performance of the estimators of the parameters. In the sequel, the problem of one-sample and two-sample prediction is discussed. A real example is given to illustrate the application of the theoretical methods given in the paper. In addition, the problem of expected experimentation time is studied with the help of drawing plots. The paper ends with several conclusions.


Main Subjects

[1] Balakrishnan, N. and Aggarwala, R., Progressive censoring: theory, methods, and applications, Birkhäuser, Boston, 2000.
[2] Basak, I., Basak, P. and Balakrishnan, N., On some predictors of times to failure of censored items in progressively censored samples, Comput. Statist. Data Anal., 50 (2006), 1313–1337.
[3] Calabria, R. and Pulcini, G., An engineering approach to Bayes estimation for the Weibull distribution, Microelectron. Reliab., 34 (1994), 789–802.
[4] Calabria, R. and Pulcini, G., Point estimation under asymmetric loss functions for left-truncated exponential samples, Comm. Statist. Theory Methods, 25 (1996), 585–600.
[5] Chen, M. H. and Shao, Q. M., Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist., 8 (1999), 69–92.
[6] Chivers, C., MHadaptive: General Markov chain Monte Carlo for Bayesian inference using adaptive Metropolis-Hastings sampling, R package version 1.1-8, (2015), ”https://CRAN.R-project.org/package=MHadaptive”.
[7] Dey, S. and Dey, T., Statistical inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal, Appl. Math. Model., 38 (2014), 974–982.
[8] Dey, S., Kayal, T. and Tripathi, Y. M., Statistical inference for the weighted exponential distribution under progressive Type-II censoring with binomial removal, Amer. J. Math. Management Sci., 37 (2018), 188–208.
[9] Geweke, J. Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, In Bayesian Statistics 4, Eds. J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, Clarendon Press, Oxford, UK, (1992), pp. 169–193.
[10] Ghitany, M. E., Al-Mutairi, D. K. and Aboukhamseen, S. M., Estimation of the reliability of a stressstrength system from power Lindley distributions, Comm. Statist. Simulation Comput., 44 (2015), 118–136.
[11] Ghitany, M. E., Al-Mutairi, D. K., Balakrishnan, N. and Al-Enezi, L. J., Power Lindley distribution and associated inference, Comput. Statist. Data Anal., 64 (2013), 20–33.
[12] Hasselman, B., nleqslv: Solve systems of nonlinear equations, R package version 3.3.2, (2018), ”https://CRAN.R-project.org/package=nleqslv”.
[13] Heidelberger, P. and Welch, P. D., A Spectral method for confidence interval generation and run length control in simulations, Commun. ACM, 24 (1981), 233–245.
[14] Heidelberger, P. and Welch, P. D., Adaptive spectral methods for simulation output analysis, IBM J. Res. Dev., 25 (1981), 860–876.
[15] Heidelberger, P. and Welch, P. D., Simulation run length control in the presence of an initial transient, Oper. Res., 31 (1983), 1109–1144.
[16] Joukar, A., Ramezani, M. and MirMostafaee, S. M. T. K., Estimation of P(X > Y ) for the power Lindley distribution based on progressively type II right censored samples, J. Stat. Comput. Simul., 90 (2020), 355–389.
[17] Kerman, J. Neutral noninformative and informative conjugate beta and gamma prior distributions, Electron. J. Stat., 5 (2011), 1450–1470.
[18] Marinho, P. R. D., Bourguignon, M. and Dias, C. R. B., AdequacyModel: Adequacy of probabilistic models and general purpose optimization, R package version 2.0.0., (2016), ”https:// CRAN.Rproject.org/package=AdequacyModel”.
[19] Pak, A. and Dey, S., Statistical inference for the power Lindley model based on record values and inter-record times, J. Comput. Appl. Math., 347 (2019), 156–172.
[20] Plummer, M., Best, N., Cowles, K. and Vines, K., CODA: Convergence diagnosis and output analysis for MCMC, R News, 6 (2006), 7–11.
[21] Plummer, M., Best, N., Cowles, K., Vines, K., Sarkar, D., Bates, D., Almond, R. and Magnusson, A., coda: Output analysis and diagnostics for MCMC, R package version 0.19-4, (2020), ”https://CRAN.R-project.org/package=coda”.
[22] Proschan, F., Theoretical explanation of observed decreasing failure rate, Technometrics, 5 (1963), 375–383.
[23] R Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2020.
[24] Raftery, A. E. and Lewis, S. M., Comment: One long run with diagnostics: Implementation strategies for Markov chain Monte Carlo, Statist. Sci., 7 (1992), 493–497.
[25] Raftery, A. E. and Lewis, S. M., Implementing MCMC, In Markov chain Monte Carlo in practice, Eds. W.R. Gilks, S. Richardson, D.J. Spiegelhalter, Chapman and Hall/CRC, Boca Raton, (1996), pp. 115–130.
[26] Ripley, B., Venables, B., Bates, D. M., Hornik, K., Gebhardt, A. and Firth, D., MASS: Support functions and datasets for Venables and Ripley’s MASS, R package version 7.3-53, (2020), ”https://CRAN.R-project.org/package=MASS”.
[27] Robert, C. P. and Casella, G., Monte Carlo statistical methods, 2nd Ed., Springer-Verlag, New York, 2004.
[28] Schruben, L. W., Detecting initialization bias in simulation output, Oper. Res., 30 (1982), 569–590.
[29] Schruben, L. Singh, H. and Tierney, L., A test of initialization bias hypotheses in simulation output, Technical Report 471, School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, 1980.
[30] Schruben, L. Singh, H. and Tierney, L., Optimal tests for initialization bias in simulation output, Oper. Res., 31 (1983), 1167–1178.
[31] Shao, J., Mathematical statistics, 2nd Ed., Springer-Verlag, New York, 2003.
[32] Singh, S. K., Singh, U. and Kumar, M., Bayesian estimation for Poisson-exponential model under progressive type-II censoring data with binomial removal and its application to ovarian cancer data, Comm. Statist. Simulation Comput., 45 (2016), 3457–3475.
[33] Singh, S. K., Singh, U. and Sharma, V. K., Expected total test time and Bayesian estimation for generalized Lindley distribution under progressively Type-II censored sample where removals follow the Beta-binomial probability law, Appl. Math. Comput., 222 (2013), 402–419.
[34] Tse, S. K. and Xiang, L., Interval estimation for Weibull-distributed life data under Type II progressive censoring with random removals, J. Biopharm. Statist., 13 (2003), 1–16.
[35] Tse, S. K., Yang, C. and Yuen, H. K., Statistical analysis of Weibull distributed lifetime data under Type II progressive censoring with binomial removals, J. Appl. Stat., 27 (2000), 1033–1043.
[36] Valiollahi, R., Raqab, M. Z., Asgharzadeh, A. and Alqallaf, F. A., Estimation and prediction for power Lindley distribution under progressively type II right censored samples, Math. Comput. Simulation, 149 (2018), 32–47.
[37] Venables, W. N. and Ripley, B. D., Modern applied statistics with S, 4th Ed., Springer-Verlag, New York, 2002.
[38] Wu, C. C., Wu, S. F. and Chan, H. Y. MLE and the estimated expected test time for the two-parameter Gompertz distribution under progressive censoring with binomial removals, Appl. Math. Comput., 181 (2006), 1657–1670.
[39] Yuen, H. K. and Tse, S. K., Parameters estimation for Weibull distributed lifetimes under progressive censoring with random removals, J. Stat. Comput. Simul., 55 (1996), 57–71.