Estimation for the Three-Parameter Exponentiated Weibull Distribution under Progressive Censored Dat

Document Type : Original Article


1 Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Iran.

2 Department of Mathematics and Statistics, Lahijan Branch, Islamic Azad University, Lahijan, Iran.


In this paper, we consider the problem of estimating the unknown parameters of an exponentiated Weibull distribution when the data are observed in the presence of progressively Type II censoring. We observed that the maximum likelihood estimators do not have a closed form, and so require a numerical technique to compute, further the implementation of the EM algorithm still requires the numerical techniques. So we employ the stochastic expectation-maximization (SEM) algorithm to estimate the model parameters and further to construct the associated asymptotic confidence intervals of the unknown parameters. Moreover, under Bayesian approach, we consider symmetric and asymmetric loss functions and compute the Bayesian estimates using the Lindley’s approximation and Gibbs sampler together with Metropolis Hastings algorithm. The highest posterior density (HPD) credible intervals are also constructed. The behavior of suggested estimators is assessed using a simulation study. Finally, a real life example is considered to illustrate the application and development of the inference methods.


Balakrishnan, N., and Cohen, A. C., (2014), Order statistics–inference: estimation methods, Elsevier, Amsterdam.
Balakrishnan, N., and Sandhu, R. A., (1995), A simple simulation algorithm for generating progressive type-II censored samplesy, American Statistician, 49(2), 229-230.
Baratpour, S., Habibirad, A., (2016), Exponentiality Test Based on the Progressive Type II Censoring via Cumulative Entropy, Journal Communications in Statistics - Simulation and Computation, 45(7), 2625-2637.
Belaghi, R.A., Asl, M.N., Alma, O.G., Singh, S., Vasfi; M., (2019), Estimation and prediction for the Poisson-Exponential distribution based on type-II censored data, American Journal of Mathematical and Management Sciences, 38(1), 96-11.
Childs, A., Chandrasekar, B., Balakrishnan, N., and Kundu, D., (2003), Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Annals of the Institute of Statistical Mathematics, 55, 319-330.
Chung, Y., Jung, M., (2013), Bayesian Inference of Three-parameter Exponentiated Weibull Distribution Based on Progressive Type-II Censoring, Journal of the Korean Data Analysis Society, 15, 585-591.
Dempster, A. P., Laird, N. M., and Rubin, D. B., (1977), Maximum likelihood from incomplete data via the EM algorithm. Journal of Royal Statistical Society, Series B, 39, 1-38.
Epstein, B., (1954), Truncated life-test in exponential case. Annals of Mathematical Statistics, 25, 555-564.
Gamchi, F.V., Alma, O. G., and Belaghi, R. A., (2019), Classical and Bayesian inference for Burr Type III distribution based on progressive Type II hybrid censored data. Mathematical Sciences, 13, 79-95.
Gumbel, E. J., (1973), Statistics of Extremes. published Colombia University Press,New York.
Gupta, R. D., and Kundu, D., (2001), Exponentiated Exponential Family: AnAlternative to Gamma andWeibull Distributions. Biometrical Journal, 43, 117-130.
Habibirad, A., Yousefzadeh, F., Balakrishnan, N., (2011), Goodness-of-fit Test Based on Kullback-Leibler Information for Progressively Type-II Censored Data. IEEE Transactions on Reliability, 60(9), 570-579.
Habibi Rad, A., and Izanlo, M., (2012), AnEMAlgorithm for Estimating the Parameters of the Generalized Exponential Distribution under Unified Hybrid Censored Data. Journal of Statistical Research of Iran, 8, 215-228.
Hastings,W. K., (1970), Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97-109.
Helu, A., Samawi, H., and Raqab, M. Z.,(2015), Estimation on Lomax progressive censoring using the EM algorithm. Journal of Statistical Computation and Simulation, 85(11), 1035-1052.
Huang, S. R., and Wu, S. J., (2012), Bayesian estimation and prediction for Weibull model with progressive censoring. Journal of Statistical Computation and Simulation, 82(11), 1607-1620.
Kim, C., Jung, J., and Chung, Y., (2011), Bayesian estimation for the exponentiated Weibull model under type II progressive censoring. Statistical Papers, 52,2030-2041.
Kumar, D., Saran, J., and Jain, N., (2017), The exponentiated burr xii distribution: moments and estimation based on lower record values. Sri Lankan Journal of Applied Statistics, 18,1-17.
Lindley, D.V., (1980), Approximate Bayesian method. Trabajos de Estadistica, 31, 223-245.
Louis, T. A., (1982), Finding the observed information matrix using the EM algorithm. Journal of Royal Statistical Society,Series B, 44, 226-233.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., (1953), Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087-1092.
Nichols, M. D., Padgett,W. J., (2006), A bootstrap control chart forWeibull percentiles. Quality and Reliability Engineering Internationa, 22, 141-151.
Panahi, H., (2018), Inference for exponentiated Pareto distribution based on progressive first-failure censored data with application to cumin essential oil data. Journal of Statistics and Management Systems, 21, 1433-1457.
Panahi, H., (2020), Interval Estimation of Kumaraswamy Parameters Based on Progressively Type II Censored Sample and Record Values. Miskolc Mathematical Notes, 21, 319-334.
Panahi, H., Moradi, N., (2020), Estimation of the inverted exponentiated Rayleigh Distribution Based on Adaptive Type II Progressive Hybrid Censored Sample. Journal of Computational and Applied Mathematicsn, 364,112345.
Sawadogo, I., Odongo, L., and Ly, I., (2017), Maximum Likelihood Estimation of the Parameters of exponentiated generalized Weibull based on progressive Type-II censored data. Open Journal of Statisticsn, 7, 956-963.
Singh, S. K., Singh, U., and Kumar, D., (2012), Bayes estimators of the reliability function and parameters of inverted exponential distribution using informative and noninformative priors. Journal of Statistical computation and simulation, 83(12), 2258-2269.
Singh, S., and Tripathi, Y. M., (2018), Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring. Statistical Papers, 59(1), 21-56.
Singh, S., Belagh, R.A., Asl; M.N. (2019), Estimation and prediction using classical and Bayesian approaches for Burr III model under progressive type-I hybrid censoring. International Journal of System Assurance Engineering and Management, 10(4),746-764.
Sobhi, M. M. A., and Soliman, A. A., (2016), Estimation for the exponentiated Weibull model with adaptive Type-II progressive censored scheme. Applied Mathematical Modelling, 40(2),1180-1192.
Tian, Y., Zhu, Q., and Tian, M., ((2015), Estimation for mixed exponential distributions under type-II progressively hybrid censored samples. Computational Statistics and Data Analysis, 89, 85-96.
Wang, L., (2018), Inference of progressively censored competing risks data from Kumaraswamy distributions. Journal of Computational and Applied Mathematics, 343(3), 719-736.
Volume 21, Issue 1
June 2022
Pages 153-177
  • Receive Date: 04 November 2020
  • Revise Date: 17 December 2021
  • Accept Date: 19 May 2022