Estimation and Prediction for Type-II Hybrid Censored Data Follow Flexible Weibull Distribution


  • Vikas Kumar Sharma Institute of Infrastructure Technology, Research and Management (IITRAM)



Flexible Weibull distribution, Type-II hybrid censored sample, Maximum likelihood estimation, Bayesian estimation, Bayesian prediction


In this paper, we proposed Bayes estimators for estimating the parameters, reliability, hazard rate, mean time to failure from flexible Weibull distribution using Type-II hybrid censored sample. Bayes estimators have been obtained under squared error loss function assuming independent gamma prior distributions for the parameters. The maximum likelihood estimators along with asymptotic distributions have also been discussed. The performances of the estimators have been compared with respect to the various Type-II hybrid censoring schemes. For approximating the posteriors, we proposed the use of Markov chain Monte Carlo techniques such as Gibbs sampler and Metropolis-Hastings algorithm. Further, Bayesian One- andTwo-sample prediction problems have also been considered. A real data set has been analysed for illustration purposes.


W. M. AFIFY (2016). On estimating flexible Weibull parameters with Type-I progressive interval censoring with random removal using data of cancerous tumors in blood. Biometrics & Biostatistics International Journal, 4, pp. 1–9.

B. AL-ZAHRANI, M. GINDWAN (2014). Parameter estimation of a two-parameter Lindley distribution under hybrid censoring. International Journal of System Assurance Engineering and Management, 5, pp. 628–636.

N. BALAKRISHNAN, D. KUNDU (2013). Hybrid censoring: Models, inferential results and applications. Computational Statistics and Data Analysis, 57, pp. 166–209.

N. BALAKRISHNAN, R. A. SHAFAY (2012). One- and two-sample Bayesian prediction intervals based on Type-II hybrid censored data. Communications in Statistics - Theory and Methods, 41, pp. 1511–1531.

A. BANERJEE, D. KUNDU (2008). Inference based on Type-II hybrid censored data from a Weibull distribution. IEEE Transaction on Reliability, 57, pp. 369–378.

M. BEBBINGTON, C.-D. LAI, R. ZITIKIS (2007). A flexible Weibull extension. Reliability Engineering and System Safety, 92, pp. 719–726.

J. O. BERGER (1985). Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York.

S. BROOKS (1998). Markov chain Monte Carlo method and its application. Journal of the Royal Statistical Society: Series D, 47, pp. 69–100.

M. CHEN, Q. SHAO (1998). Monte carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 6, pp. 66–92.

S. M. CHEN, G. K. BHATTACHARYYA (1988). Exact confidence bound for an exponential under hybrid censoring. Communication in Statistics - Theory and Methods, 17, pp. 1858–1870.

A. CHILDS, B. CHANDRASEKAR, N. BALAKRISHNAN, D. KUNDU (2003). Exact likelihood inference based on type-I and type-II hybrid censored sample from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, pp. 319–330.

I. R. DANSMORE (1974). The Bayesian predictive distribution in life testing models. Technometrics, 16, pp. 455–460.

S.DUBEY, B. PRADHAN, D.KUNDU (2011). Parameter estimation of the hybrid censored log-normal distribution. Journal of Statistical Computation and Simulation, 81, pp. 275–287.

N. EBRAHMINI (1992). Prediction intervals for future failures in the exponential distribution under hybrid censoring. IEEE Transaction on Reliability, 41, pp. 127–132.

B. EFRON, R. TIBSHIRANI (1986). Bootstrap methods for standars errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, 1, pp. 54–75.

B. EPSTEIN (1954). Truncated life test in the exponential case. Annals of Mathematical Statistics, 25, pp. 555–564.

A. GANGULY, S. MITRA, D. SAMANTA, D. KUNDU (2012). Exact inference for the two parameter exponential distribution under Type-II hybrid censoring. Journal of Statistical Planning and Inference, 142, pp. 613–625.

P. K. GUPTA, B. SINGH (2012). Parameter estimation of Lindley distribution with hybrid censored data. International Journal of System Assurance Engineering and Management, 1, pp. 1–8.

W. HASTINGS (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 55, pp. 97–109.

H. JEFFREY (1961). Theory of Probability. Oxford University Press, Oxford, UK, 3rd ed.

A. KOHANSAL, S. REZAKHAH, E. KHORRAM (2015). Parameter estimation of Type-II hybrid censored weighted exponential distribution. Communications in Statistics - Simulation and Computation, 44, pp. 1273–1299.

D. KUNDU (2007). On hybrid censored Weibull distribution. Journal of Statistical Planning and Inference, 137, pp. 2127–2142.

J. F. LAWLESS (1971). A prediction problem concerning samples from the exponential distribution, with application in life testing. Technometrics, 13, pp. 725–730.

MIL-STD-781C (1977). Reliability Design Qualification and Production Acceptance Test, Exponential Distribution. U.S. Government printing office,Washington, DC.

S. PARK, N. BALAKIRSHNAN (2012). A very flexible hybrid censoring scheme and its Fisher information. Journal of Statistical Computation and Simulation, 82, pp. 41–50.

M. K. RASTOGI, Y. M. TRIPATHI (2012). Inference on unknown parameters of a Burr distribution under hybrid censoring. Statistical Papers, 53, pp. 1–25.

M. A. SELIM, H. M. SALEM (2014). Recurrence relations for moments of k-th upper record values from flexible Weibull distribution and a characterization. American Journal of Applied Mathematics and Statistics, 2, pp. 168–171.

R. A. SHAFAY, N. BALAKRISHNAN (2012). One and two sample Bayesian prediction intervals based on Type-I hybrid censored data. Communication in Statistics - Simulation and Computation, 41, pp. 65–88.

S. K. SINGH, U. SINGH, V. K. SHARMA (2013). Bayesian estimation and prediction for flexible Weibull model under Type-II censoring scheme. Journal of Probability and Statistics, 2013, pp. 1–16.

S. K. SINGH, U. SINGH, V. K. SHARMA (2016). Estimation and prediction for Type-I hybrid censored data from generalized Lindley distribution. Journal of Statistics and Management Systems, 19, pp. 367–396.

S. K. SINGH, U. SINGH, V. K. SHARMA, M. KUMAR (2015). Estimation for flexible Weibull extension under progressive Type-II censoring. Journal of Data Science, 13, pp. 21–42.

A. SMITH, G. ROBERTS (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society: Series B, 55, pp. 3–23.

M. S. SUPRAWHARDANA, PRAYOTO, SANGADJI (1999). Total time on test plot analysis for mechanical components of the RSG-GAS reactor. Atom Indonesia, 25, pp. 81-90.

A. S. YADAV, S. K. SINGH, U. SINGH (2016). On hybrid censored inverse Lomax distribution: Application to the survival data. Statistica, 76, pp. 185–203.




How to Cite

Sharma, V. K. (2018). Estimation and Prediction for Type-II Hybrid Censored Data Follow Flexible Weibull Distribution. Statistica, 77(4), 385–414.