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Hybrid censoring: models, inferential results and applications. (English) Zbl 1365.62364
Summary: A hybrid censoring scheme is a mixture of Type-I and Type-II censoring schemes. In this review, we first discuss Type-I and Type-II hybrid censoring schemes and associated inferential issues. Next, we present details on developments regarding generalized hybrid censoring and unified hybrid censoring schemes that have been introduced in the literature. Hybrid censoring schemes have been adopted in competing risks set-up and in step-stress modeling and these results are outlined next. Recently, two new censoring schemes, viz., progressive hybrid censoring and adaptive progressive censoring schemes have been introduced in the literature. We discuss these censoring schemes and describe inferential methods based on them, and point out their advantages and disadvantages. Determining an optimal hybrid censoring scheme is an important design problem, and we shed some light on this issue as well. Finally, we present some examples to illustrate some of the results described here. Throughout the article, we mention some open problems and suggest some possible future work for the benefit of readers interested in this area of research.

##### MSC:
 62N01 Censored data models 62N02 Estimation in survival analysis and censored data 62N05 Reliability and life testing 62F10 Point estimation
SPLIDA
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##### References:
 [1] AL-Hussaini, E. K., Predicting observables from a general class of distributions, Journal of Statistical Planning and Inference, 79, 79-91, (1999) · Zbl 0933.62018 [2] Bagdonavicius, V.; Nikulin, M., Accelerated life models: modeling and statistical analysis, (2002), Chapman & Hall, CRC Press Boca Raton, FL · Zbl 1001.62035 [3] Balakrishnan, N., Progressive censoring methodology: an appraisal (with discussions), TEST, 16, 211-296, (2007) · Zbl 1121.62052 [4] Balakrishnan, N., A synthesis of exact inferential results for exponential step-stress models and associated optimal accelerated life-tests, Metrika, 69, 351-396, (2009) · Zbl 1433.62287 [5] Balakrishnan, N.; Aggarwala, R., Progressive censoring: theory, methods, and applications, (2000), Birkhäuser Boston, MA [6] Balakrishnan, N.; Cohen, A. C., Order statistics and inference: estimation methods, (1991), Academic Press San Diego, CA · Zbl 0732.62044 [7] Balakrishnan, N.; Habibi Rad, A.; Arghami, N. R., Testing exponentiality based on kullback – leibler information with progressively type-II censored data, IEEE Transactions on Reliability, 56, 301-307, (2007) [8] Balakrishnan, N.; Iliopoulos, G., Stochastic monotonicity of the MLE of exponential mean under different censoring schemes, Annals of the Institute of Statistical Mathematics, 61, 753-772, (2009) · Zbl 1332.62384 [9] Balakrishnan, N.; Iliopoulos, G., Stochastic monotonicity of the MLEs of parameters in exponential simple step-stress models under type-I and type-II censoring, Metrika, 72, 89-109, (2010) · Zbl 1189.62160 [10] Balakrishnan, N.; Kannan, N.; Lin, C.-T.; Wu, S. J.S., Inference for the extreme value distribution under progressive type-II censoring, Journal of Statistical Computation and Simulation, 74, 25-45, (2004) · Zbl 1048.62090 [11] Balakrishnan, N.; Kateri, M., On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data, Statistics & Probability Letters, 78, 2971-2975, (2008) · Zbl 1148.62081 [12] Balakrishnan, N.; Kundu, D.; Ng, H. K.T.; Kannan, N., Point and interval estimation for a simple step-stress model with type-II censoring, Journal of Quality Technology, 39, 35-47, (2007) [13] Balakrishnan, N.; Lin, C.-T., On the distribution of a test for exponentiality based on progressively type-II right censored spacings, Journal of Statistical Computation and Simulation, 73, 277-283, (2003) · Zbl 1052.62014 [14] Balakrishnan, N.; Ng, H. K.T.; Kannan, N., Goodness-of-fit tests based on spacings for progressively type-II censored data from a general location-scale distribution, IEEE Transactions on Reliability, 53, 349-356, (2004) [15] Balakrishnan, N.; Rasouli, A.; Farsipour, N. S., Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution, Journal of Statistical Computation and Simulation, 78, 475-488, (2008) · Zbl 1274.62667 [16] Balakrishnan, N.; Shafay, A. R., One- and two-sample Bayesian prediction intervals based on type-II hybrid censored data, Communications in Statistics—Theory and Methods, 41, 1511-1531, (2012) · Zbl 1319.62198 [17] Balakrishnan, N.; Varadhan, J., Approximate MLEs for the location & scale parameters of the extreme value distribution with censoring, IEEE Transactions on Reliability, 40, 146-151, (1991) · Zbl 0729.62538 [18] Balakrishnan, N.; Xie, Q., Exact inference for a simple step-stress model with type-I hybrid censored data from the exponential distribution, Journal of Statistical Planning and Inference, 137, 3268-3290, (2007) · Zbl 1119.62096 [19] Balakrishnan, N.; Xie, Q., Exact inference for a simple step-stress model with type-II hybrid censored data from the exponential distribution, Journal of Statistical Planning and Inference, 137, 2543-2563, (2007) · Zbl 1115.62109 [20] Balakrishnan, N.; Xie, Q.; Kundu, D., Exact inference for a simple step-stress model from the exponential distribution under time constraint, Annals of the Institute of Statistical Mathematics, 61, 251-274, (2009) · Zbl 1294.62233 [21] Balasooriya, U.; Balakrishnan, N., Reliability sampling plans for log-normal distribution based on progressively censored samples, IEEE Transactions on Reliability, 49, 199-203, (2000) [22] Banerjee, A.; Kundu, D., Inference based on type-II hybrid censored data from a Weibull distribution, IEEE Transactions on Reliability, 57, 369-378, (2008) [23] Barlow, R. E.; Madansky, A.; Proschan, F.; Scheuer, E., Statistical estimation procedures for the burn-in process, Technometrics, 10, 51-62, (1968) [24] Bartholomew, D. J., The sampling distribution of an estimate arising in life testing, Technometrics, 5, 361-374, (1963) · Zbl 0121.14403 [25] Berger, J. O.; Sun, D., Bayesian analysis for the poly-Weibull distribution, Journal of the American Statistical Association, 88, 1412-1418, (1993) · Zbl 0792.62020 [26] Chandrasekar, B.; Childs, A.; Balakrishnan, N., Exact likelihood inference for the exponential distribution under generalized type-I and type-II hybrid censoring, Naval Research Logistics, 51, 994-1004, (2004) · Zbl 1162.62317 [27] Chen, S.; Bhattacharyya, G. K., Exact confidence bounds for an exponential parameter under hybrid censoring, Communications in Statistics—Theory and Methods, 17, 1857-1870, (1988) · Zbl 0644.62101 [28] Childs, A., Balakrishnan, N., Chandrasekar, B., 2012. Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring. Statistics (in press). · Zbl 1314.62050 [29] Childs, A.; Chandrasekar, B.; Balakrishnan, N., Exact likelihood inference for an exponential parameter under progressive hybrid censoring, (Vonta, F.; Nikulin, M.; Limnios, N.; Huber-Carol, C., Statistical Models and Methods for Biomedical and Technical Systems, (2008), Birkhäuser Boston, MA), 319-330 · Zbl 1049.62021 [30] Childs, A.; Chandrasekar, B.; Balakrishnan, N.; Kundu, D., 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, (2003) · Zbl 1049.62021 [31] Cox, D. R., The analysis of exponentially distributed lifetimes with two types of failure, Journal of the Royal Statistical Society, Series B, 21, 411-421, (1959) · Zbl 0093.15704 [32] Crowder, M., Classical competing risks, (2001), Chapman and Hall Boca Raton, FL · Zbl 0979.62078 [33] (D’Agostino, R. B.; Stephens, M. A., Goodness-of-Fit Techniques, (1986), Marcel Dekker New York, NY) [34] Das, B.; Nag, A. S., A test of exponentiality in life-testing against Weibull alternatives under hybrid censoring, Calcutta Statistical Association Bulletin, 52, 371-380, (2002) [35] Devroye, L., A simple algorithm for generating random samples from a log-concave density function, Computing, 33, 247-257, (1984) · Zbl 0561.65004 [36] Draper, N.; Guttman, I., Bayesian analysis of hybrid life tests with exponential failure times, Annals of the Institute of Statistical Mathematics, 39, 219-225, (1987) · Zbl 0612.62134 [37] Dube, S.; Pradhan, B.; Kundu, D., Parameter estimation of the hybrid censored log-normal distribution, Journal of Statistical Computation and Simulation, 81, 275-287, (2011) · Zbl 1221.62137 [38] Epstein, B., Truncated life-test in exponential case, Annals of Mathematical Statistics, 25, 555-564, (1954) · Zbl 0058.35104 [39] Fairbanks, K.; Madsan, R.; Dykstra, R., A confidence interval for an exponential parameter from hybrid life-test, Journal of the American Statistical Association, 77, 137-140, (1982) · Zbl 0504.62087 [40] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721-741, (1984) · Zbl 0573.62030 [41] Gupta, R. D.; Kundu, D., Hybrid censoring schemes with exponential failure distribution, Communications in Statistics—Theory and Methods, 27, 3065-3083, (1998) · Zbl 1008.62679 [42] Gupta, R. D.; Kundu, D., Generalized exponential distributions, Australian and New Zealand Journal of Statistics, 41, 173-188, (1999) · Zbl 1007.62503 [43] Gupta, R. D.; Kundu, D., On the comparison of Fisher information matrices of the Weibull and generalized exponential distributions, Journal of Statistical Planning and Inference, 136, 3130-3144, (2006) · Zbl 1094.62122 [44] Gupta, R. D.; Kundu, D., Generalized exponential distributions: existing methods and recent developments, Journal of Statistical Planning and Inference, 137, 3537-3547, (2007) · Zbl 1119.62011 [45] Habibi Rad, A.; Yousefzadeh, F.; Balakrishnan, N., Goodness-of-fit test based on kullback – leibler information for progressively type-II censored data, IEEE Transactions on Reliability, 60, 570-579, (2011) [46] Huang, W. T.; Lin, Y. P., An improved Bayesian sampling plan for exponential population with type I censoring, Communications in Statistics—Theory and Methods, 31, 2003-2025, (2002) · Zbl 1051.62120 [47] Jeong, H.-S.; Park, J.-I.; Yum, B.-J., Development of $$(r, T)$$ hybrid sampling plans for exponential lifetime distributions, Journal of Applied Statistics, 23, 601-607, (1996) [48] Kaminskiy, M. P.; Krivtsov, V. V., A simple procedure for Bayesian estimation of the Weibull distribution, IEEE Transactions on Reliability, 54, 612-616, (2005) [49] Kateri, M.; Balakrishnan, N., Inference for a simple step-stress model with type-II censoring, and Weibull distributed lifetimes, IEEE Transactions on Reliability, 57, 616-626, (2008) [50] Kundu, D., On hybrid censored Weibull distribution, Journal of Statistical Planning and Inference, 137, 2127-2142, (2007) · Zbl 1120.62081 [51] Kundu, D., Bayesian inference and life testing plan for Weibull distribution in presence of progressive censoring, Technometrics, 50, 144-154, (2008) [52] Kundu, D.; Gupta, R. D., Analysis of hybrid lifetests in presence of competing risks, Metrika, 65, 159-170, (2007) · Zbl 1106.62111 [53] Kundu, D.; Joarder, A., Analysis of type-II progressively hybrid censored data, Computational Statistics & Data Analysis, 50, 2509-2528, (2006) · Zbl 1284.62605 [54] Kundu, D.; Pradhan, B., Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring, Communications in Statistics—Theory and Methods, 38, 2030-2041, (2009) · Zbl 1167.62078 [55] Kundu, D., Samanta, D., Ganguli, A., Mitra, S., 2011. Bayesian analysis of different hybrid and progressive life tests (submitted for publication). [56] Liang, T.-C., Yang, M.-C., 2012. Optimal Bayes sampling plans for exponential distributions based on hybrid censored samples, Journal of Statistical Computation and Simulation (in press). [57] Lin, C.-T.; Huang, Y.-L.; Balakrishnan, N., Exact Bayesian variable sampling plans for exponential distribution under type-I censoring, (Huber, C.; Limnios, N.; Mesbah, M.; Nikulin, M., Mathematical Methods for Survival Analysis, Reliability and Quality of Life, (2007), Hermes London, UK), 155-166 [58] Lin, C.-T.; Huang, Y.-L.; Balakrishnan, N., Exact Bayesian variable sampling plans for the exponential distribution based on type-I and type-II hybrid censored samples, Communications in Statistics—Simulation and Computation, 37, 1101-1116, (2008), Corrections, vol. 39, 1499-1505 · Zbl 1145.62004 [59] Lin, C-T.; Huang, Y-L.; Balakrishnan, N., A new method for goodness-of-fit testing based on type-II right censored samples, IEEE Transactions on Reliability, 57, 633-642, (2008) [60] Lin, C-T.; Huang, Y-L.; Balakrishnan, N., Exact Bayesian variable sampling plans for the exponential distribution based on progressive hybrid censoring, Journal of Statistical Computation and Simulation, 81, 873-882, (2011) · Zbl 1219.62010 [61] Lin, C-T.; Ng, H. K.T.; Chan, P. S., Statistical inference of type-II progressively hybrid censored data with Weibull lifetimes, Communications in Statistics—Theory and Methods, 38, 1710-1729, (2009) · Zbl 1165.62018 [62] Lin, Y-P.; Liang, T.; Huang, W-T., Bayesian sampling plans for exponential distribution based on type-I censoring, Annals of the Institute of Statistical Mathematics, 54, 100-113, (2002) · Zbl 0993.62099 [63] Meeker, W. Q.; Escobar, L. A., Statistical models for reliability data, (1998), John Wiley & Sons New York, NY · Zbl 0949.62086 [64] MIL-STD-781-C, 1977. Reliability design qualification and production acceptance tests: exponential distribution, US Government Printing Office, Washington, DC. [65] Mokhtari, E. B.; Habibi Rad, A.; Yousefzadeh, F., Inference for Weibull distribution based on progressively type-II hybrid censored data, Journal of Statistical Planning and Inference, 141, 2824-2838, (2011) · Zbl 1213.62034 [66] Mudholkar, G. S.; Srivastava, D. K., Exponentiated Weibull family for analyzing bathtub failure data, IEEE Transactions on Reliability, 42, 299-302, (1993) · Zbl 0800.62609 [67] Nelson, W., Accelerated life testing: step-stress models and data analysis, IEEE Transactions on Reliability, 29, 103-108, (1980) · Zbl 0462.62078 [68] Nelson, W., Applied life data analysis, (1982), John Wiley & Sons New York, NY · Zbl 0579.62089 [69] Nelson, W., Accelerated testing: statistical models, test plans and data analysis, (1990), John Wiley & Sons New York, NY [70] Ng, H. K.T.; Chan, P. S.; Balakrishnan, N., Estimation of parameters from progressively censored data using EM algorithm, Computational Statistics & Data Analysis, 39, 371-386, (2002) · Zbl 0993.62085 [71] Ng, H. K.T.; Chan, P. S.; Balakrishnan, N., Optimal progressive censoring plans for the Weibull distribution, Technometrics, 46, 470-481, (2004) [72] Ng, H. K.T.; Kundu, D.; Chan, P. S., Statistical analysis of exponential lifetimes under an adaptive hybrid type-II progressive censoring scheme, Naval Research Logistics, 56, 687-698, (2009) · Zbl 1178.62111 [73] Pakyari, R.; Balakrishnan, N., A general purpose approximate goodness-of-fit test for progressively type-II censored data, IEEE Transactions on Reliability, 61, 238-244, (2012) [74] Pakyari, R., Balakrishnan, N., 2012b. Goodness-of-fit tests for progressively Type-II censored data from location-scale distributions, Journal of Statistical Computation and Simulation (in press). · Zbl 1348.62163 [75] Park, S.; Balakrishnan, N., On simple calculation of the Fisher information in hybrid censoring schemes, Statistics & Probability Letters, 79, 1311-1319, (2009) · Zbl 1162.62091 [76] Park, S.; Balakrishnan, N., A very flexible hybrid censoring scheme and its Fisher information, Journal of Statistical Computation and Simulation, 82, 41-50, (2012) · Zbl 1431.62439 [77] Park, S.; Balakrishnan, N.; Kim, S. W., Fisher information in progressive hybrid censoring schemes, Statistics, 45, 623-631, (2011) · Zbl 1284.62616 [78] Park, S.; Balakrishnan, N.; Zheng, G., Fisher information in hybrid censored data, Statistics & Probability Letters, 78, 2781-2786, (2008) · Zbl 1154.62072 [79] Pintilie, M., Competing risks: A practical perspective, (2006), John Wiley & Sons Hoboken, NJ · Zbl 1120.62076 [80] Raqab, M. Z.; Madi, M. T., Bayesian inference for the generalized exponential distribution, Journal of Statistical Computation and Simulation, 75, 841-852, (2005) · Zbl 1076.62025 [81] Shafay, A. R.; Balakrishnan, N., One- and two-sample Bayesian prediction intervals based on type-I hybrid censored data, Communications in Statistics—Simulation and Computation, 41, 65-88, (2012) · Zbl 06073000 [82] Shafay, A.R., Balakrishnan, N., 2012b. One- and two-sample Bayesian prediction intervals based on generalized Type-I hybrid censored data (submitted for publication). · Zbl 06073000 [83] Soland, R., Bayesian analysis of Weibull process with unknown shape and scale parameters, IEEE Transactions on Reliability, 18, 181-184, (1969) [84] Tiku, M. L.; Tan, W. Y.; Balakrishnan, N., Robust inference, (1986), Marcel Dekker New York, NY · Zbl 0597.62017 [85] Viveros, R.; Balakrishnan, N., Interval estimation of parameters of life from progressively censored data, Technometrics, 36, 84-91, (1994) · Zbl 0800.62623 [86] Wang, Y.; He, S., Fisher information in censored data, Statistics & Probability Letters, 73, 199-206, (2005) · Zbl 1065.62166 [87] Yeh, L., Bayesian variable sampling plans for the exponential distribution with type I censoring, Annals of Statistics, 22, 696-711, (1994) · Zbl 0805.62093 [88] Zhang, Y.; Meeker, W. Q., Bayesian life test planning for the Weibull distribution with given shape parameter, Metrika, 61, 237-249, (2005) · Zbl 1079.62099
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