Parameter Estimation of Type-I and Type-II Hybrid Censored Data from the Log-Logistic Distribution

Main Article Content

Seunggeun Hyun
Jimin Lee
Robert Yearout


In experiments on product lifetime and reliability testing, there are many practical situations in which researchers terminate the experiment and report the results before all items of the experiment fail because of time or cost consideration. The most common and popular censoring schemes are type-I and type-II censoring. In type-I censoring scheme, the termination time is pre-fixed, but the number of observed failures is a random variable. However, if the mean lifetime of experimental units is somewhat larger than the pre-fixed termination time, then far fewer failures would be observed and this is a significant disadvantage on the efficiency of inferential procedures. In type-II censoring scheme, however, the number of observed failures is pre-fixed, but the experiment time is a random variable. In this case, at least pre-specified number of failure are obtained, but the termination time is clearly a disadvantage from the experimenter’s point of view. To overcome some of the drawbacks in those schemes, the hybrid censoring scheme, which is a mixture of the conventional type-I and type-II censoring schemes, has received much attention in recent years. In this paper, we consider the analysis of type-I and type-II hybrid censored data where the lifetimes of items follow two-parameter log-logistic distribution. We present the maximum likelihood estimators of unknown parameters and asymptotic confidence intervals, and a simulation study is conducted to evaluate the proposed methods.

Article Details

How to Cite
Hyun, S., Lee, J., & Yearout, R. (2016). Parameter Estimation of Type-I and Type-II Hybrid Censored Data from the Log-Logistic Distribution. Industrial and Systems Engineering Review, 4(1), 37-44.
Author Biographies

Seunggeun Hyun, University of South Carolina Upstate

Associate Professor, Division of Mathematics and Computer Science

Jimin Lee, University of North Carolina Asheville

Associate Professor, Department of Mathematics

Robert Yearout, University of North Carolina Asheville

Professor, Department of Management and Accountancy


Asgharzadeh, A., Kazemi, M., and Kus, C. (2013). Analysis of the hybrid censored data from the logistic distribution. Journal of Probability and Statistical Science, 11, 183-198.

Akhtar, M. T., & Khan, A. A. (2014). Log-logistic distribution as a reliability model: a Bayesian analysis. American Journal of Mathematics and Statistics, 4, 162-170.

Balakrishnan, N. & Kundu, D. (2013). Hybrid censoring: models, inferential results and applications (with discussion). Computational Statistics and Data Analysis, 59, 166-209.

Banerjee, A. & Kundu, D. (2008). Inference based on type-II hybrid censored data from a Weibull distribution. IEEE Transactions on Reliability, 57, 369-378.

Barlow, R., Madansky, A., Proschan, F., and Scheuer, E. (1968). Statistical estimation procedures for the “Burn-in” process, Technometrics, 10, 51-62.

Bartholomew, D. (1963). The sampling distribution of an estimate arising in life testing. Technometrics, 5, 361-374.

Cohen, A. (1963). Progressively censored samples in life testing. Technometrics, 5, 327-329.

Chandrasekar, B., Childs, A., & Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring. Naval Research Logistic. 51, 994–1004.

Chen, S., & Bhattacharya, G. K. (1988). Exact confidence bounds for an exponential parameter under hybrid censoring. Communications in Statistics-Theory and Methods, 17, 1857-1870.

Childs, A., Chandrasekhar, B., Balakrishnan, N., & Kundu, D. (2003). Exact 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.

Chiodo, E., & Mazzanti, G. (2004). The log-logistic model for reliability characterization of power system components subjected to random stress. Speedam, Capri/Italy, 239-244.

Draper, N., & Guttman, I. (1987). Bayesian analysis of hybrid life tests with exponential failure times. Annals of the Institute of Statistical Mathematics, 39, 219-225.

Dube, S., Pradhan, B., & Kundu, D. (2010). Parameter estimation of the hybrid censored log normal distribution. Journal of Statistical Computation and Simulation, 81, 275-287.

Ebrahimi, N. (1992). Prediction intervals for future failures in the exponential distribution under hybrid censoring, IEEE Transactions on Reliability, 41, 127-132.

Epstein, B. (1954). Truncated life tests in the exponential case. Annals of Mathematical Statistics, 25, 555-564.

Fairbanks, K., Masson, R., and Dykstra, R. (1982). A confidence interval for an exponential parameter from a hybrid life test. Journal of the American Statistical Association, 77, 137-140.

Hyun, S., Lee, J., and Yearout, R. (2015). Analysis of the hybrid censored log-logistic distribution. Proceedings of the 4th annual world conference of the society for industrial and systems engineering, Fort Lauderdale, FL, 188-193.

Jeong, H.S., Park, J.I., and Yum, B.J. (1996). Development f (r,T) hybrid sampling plans for exponential lifetime distributions, Journal of Applied Statistics, 23, 601-607.

Jeong, H.S., Yum, B.J., (1995). Type-I censored life plans in the exponential case. Communications in Statistics-Simulation and Computation, 24, 187–205.

Kantam, R. R., Rao, G. S., & Sriram, B. (2006). An economic reliability test plan: log-logistic distribution. Journal of Applied Statistics, 33, 291-296.

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

Kundu, D., & Joarder, A. (2006). Analysis of Type-II progressively hybrid censored data. Computational Statistics and Data Analysis, 50, 2509-2528.

Kundu, D. & Pradhan, B. (2009). Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring. Communications in Statistics-Theory and Methods, 38, 2030-2041.

Most read articles by the same author(s)