Inferences for Weibull parameters under progressively first-failure censored data with binomial random removals

  • Samir K. Ashour Department of Mathematical Statistics, Institute of Statistical Studies and Research, Cairo University, Egypt
  • Ahmed A. El-Sheikh Department of Applied Statistics and Econometrics, Institute of Statistical Studies and Research, Cairo University, Egypt
  • Ahmed Elshahhat Department of Accounting and Quantitative Information Systems, Faculty of Technology and Development, Zagazig University, Egypt
Keywords: Bayes procedure, Markov chain Monte Carlo, maximum likelihood estimation, progressive first-failure censored sampling, squared error loss function, Weibull distribution


In this paper, the Bayesian and non-Bayesian estimation of a two-parameter Weibull lifetime model in presence of progressive first-failure censored data with binomial random removals are considered. Based on the s-normal approximation to the asymptotic distribution of maximum likelihood estimators, two-sided approximate confidence intervals for the unknown parameters are constructed. Using gamma conjugate priors, several Bayes estimates and associated credible intervals are obtained relative to the squared error loss function. Proposed estimators cannot be expressed in closed forms and can be evaluated numerically by some suitable iterative procedure. A Bayesian approach is developed using Markov chain Monte Carlo techniques to generate samples from the posterior distributions and in turn computing the Bayes estimates and associated credible intervals. To analyze the performance of the proposed estimators, a Monte Carlo simulation study is conducted. Finally, a real data set is discussed for illustration purposes.


M. V. Ahmadi, and M. Doostparast, Pareto analysis for the lifetime performance index of products on the basis of progressively first-failure-censored batches under balanced symmetric and asymmetric loss functions, Journal of Applied Statistics, vol. 46, no. 7, pp. 1196–1227, 2019.

E. A. Ahmed, Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application, Journal of Applied Statistics, vol. 44, no. 9, pp. 1576–1608, 2017.

S. K. Ashour, and A. Elshahhat, Bayesian and non-Bayesian estimation for Weibull parameters based on generalized Type-II progressive hybrid censoring scheme, Pakistan Journal of Statistics and Operation Research, vol. 12, no. 2, pp. 213–226, 2016.

S. K. Ashour, A. A. El-Sheikh, and A. Elshahhat, Maximum likelihood estimation of the generalised Gompertz distribution under progressively first-failure censored sampling, South African Statistical Journal, vol. 52, no. 2, pp. 115–128, 2018.

N. Balakrishnan, and E. Cramer, The Art of Progressive Censoring. Applications to Reliability and Quality, New York, NY Birkhauser, 2014.

N. Balakrishnan, and R. A. Sandhu, A simple simulational algorithm for generating progressive Type-II censored samples, The American Statistician, vol. 49, no. 2, pp. 229–230, 1995.

U. Balasooriya, Failure-censored reliability sampling plans for the exponential distribution, Journal of Statistical Computation and Simulation, vol. 52, no. 4, pp. 337–349, 1995.

M. Chacko, and R. Mohan, Bayesian analysis of Weibull distribution based on progressive Type-II censored competing risks data with binomial removals, Computational Statistics, vol. 34, no. 1, pp. 233–252, 2019.

Y. Cho, and H. Sun, and K. Lee, Estimating the entropy of a Weibull distribution under generalized progressive hybrid censoring, Entropy, vol. 17, no. 1, pp. 102–122, 2015.

S. Dey, and T. Dey, Statistical inference for the Rayleigh distribution under progressively Type-II censoring with binomial removal, Applied Mathematical Modelling, vol. 38, no. 3, pp. 974–982, 2014.

M. Dube, H. Krishna, and R. Garg, Generalized inverted exponential distribution under progressive first-failure censoring, Journal of Statistical Computation and Simulation, vol. 86, no. 6, pp. 1095–1114, 2016.

R. D. Gupta, and D. Kundu, Exponentiated exponential family: an alternative to gamma and Weibull distributions, Biometrical Journal: Journal of Mathematical Methods in Biosciences, vol. 43, no. 1, pp. 117–130, 2001.

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, vol. 57, no. 1, pp. 97–109, 1970.

A. Henningsen, and O. Toomet, maxLik: A package for maximum likelihood estimation in R, Computational Statistics, vol. 26, no. 3, pp. 443–458, 2011.

S.-R. Huang, and S.-J. Wu, Estimation of Pareto distribution under progressive first-failure censoring with random removals, Chinese Journal of Statistics, vol. 49, no. 3, pp. 82–97, 2011.

N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, Wiley Series in Probability and Statistics, 1994.

A. Kaushik, U. Singh, and S. K. Singh, Bayesian inference for the parameters of Weibull distribution under progressive Type-I interval censored data with beta-binomial removals, Communications in Statistics-Simulation and Computation, vol. 46, no. 4, pp. 3140–3158, 2017.

J. F. Lawless, Statistical Models and Methods for Lifetime Data, John Wiley & Sons, 2011.

H. Linhart, and W. Zucchini, Model Selection, John Wiley & Sons, 1986.

H. S. Mohammed, S. F. Ateya, and E. K. AL-Hussaini, Estimation based on progressive first-failure censoring from exponentiated exponential distribution, Journal of Applied Statistics, vol. 44, no. 8, pp. 1479–1494, 2017.

M. Plummer, N. Best, K. Cowles, and K. Vines, CODA: convergence diagnosis and output analysis for MCMC, R news, vol. 6, no. 1, pp. 7–11, 2006.

A. F. M. Smith, and G. O. Roberts, Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society. Series B (Methodological), vol. 55, no. 1, pp. 3–23, 1993.

A. A. Soliman, A. H. Abd Ellah, N. A. Abou-Elheggag, and R. M. El-Sagheer, Estimation based on progressive first-failure censored sampling with binomial removals, Intelligent Information Management, vol. 5, no. 4, pp. 117–125, 2013.

S. K. Tse, C. Yang, and H.-K. Yuen, Statistical analysis of Weibull distributed lifetime data under Type-II progressive censoring with binomial removals, Journal of Applied Statistics, vol. 27, no. 8, pp. 1033–1043, 2000.

S.-J. Wu, and C.-T. Chang, Parameter estimations based on exponential progressive Type-II censored data with binomial removals,International Journal of Information and Management Sciences, vol. 13, no. 3, pp. 37–46, 2002.

S.-J. Wu, and C. Kus¸, On estimation based on progressive first-failure-censored sampling, Computational Statistics and Data Analysis, vol. 53, no. 10, pp. 3659–3670, 2009.

How to Cite
Ashour, S. K., El-Sheikh, A. A., & Elshahhat, A. (2020). Inferences for Weibull parameters under progressively first-failure censored data with binomial random removals. Statistics, Optimization & Information Computing, 9(1), 47-60.
Research Articles