Exact Inference for a New Flexible Hybrid Censoring Scheme

Research Article


We introduce a new hybrid censoring scheme called general unified (progressive) hybrid censoring and study its properties by applying the modularization technique proposed in Górny and Cramer (Metrika 81(2):173–210, 2018a). For exponentially distributed lifetimes, we illustrate that already known progressive hybrid censoring models are included as particular cases. Further, we determine the exact distribution of the maximum likelihood estimators (MLEs) for an underlying exponential and uniform distribution under general unified progressive hybrid censoring and general unified hybrid censoring, respectively. The results are applied to construct exact confidence intervals.


Hybrid censoring Progressive censoring General unified hybrid censoring General unified progressive hybrid censoring Modularization Exponential distribution Uniform distribution B-spline Gamma distribution 



We are grateful to an anonymous referee for the valuable comments and suggestions which led to both an extension and an improved presentation of the results.


  1. Balakrishnan N, Cramer E (2014) The art of progressive censoring: applications to reliability and quality. Statistics for industry and technology. Springer, New YorkCrossRefMATHGoogle Scholar
  2. Balakrishnan N, Cramer E, Kamps U (2001) Bounds for means and variances of progressive type II censored order statistics. Stat Probab Lett 54(3):301–315MathSciNetCrossRefMATHGoogle Scholar
  3. Balakrishnan N, Cramer E, Iliopoulos G (2014) On the method of pivoting the CDF for exact confidence intervals with illustration for exponential mean under life-test with time constraints. Stat Probab Lett 89:124–130MathSciNetCrossRefMATHGoogle Scholar
  4. Balakrishnan N, Iliopoulos G (2009) Stochastic monotonicity of the MLE of exponential mean under different censoring schemes. Ann Inst Stat Math 61(3):753–772MathSciNetCrossRefMATHGoogle Scholar
  5. Balakrishnan N, Kundu D (2013) Hybrid censoring: models, inferential results and applications. Comput Stat Data Anal 57(1):166–209MathSciNetCrossRefMATHGoogle Scholar
  6. Balakrishnan N, Rasouli A, Farsipour NS (2008) Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution. J Stat Comput Simul 78(5):475–488MathSciNetCrossRefMATHGoogle Scholar
  7. Burkschat M, Cramer E, Górny J (2016) Type-I censored sequential \(k\)-out-of-\(n\) systems. Appl Math Model 40:8156–8174MathSciNetCrossRefGoogle Scholar
  8. Casella G, Berger RL (2002) Statistical inference, 2nd edn. Duxbury, Pacific GroveMATHGoogle Scholar
  9. Childs A, Chandrasekar B, Balakrishnan N, Kundu D (2003) Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Ann Inst Stat Math 55(2):319–330MathSciNetMATHGoogle Scholar
  10. Childs A, Chandrasekar B, Balakrishnan N (2008) Exact likelihood inference for an exponential parameter under progressive hybrid censoring schemes. In: Vonta F, Nikulin M, Limnios N, Huber-Carol C (eds) Statistical models and methods for biomedical and technical systems, statistics for industry and technology. Birkhäuser, Boston, pp 323–334Google Scholar
  11. 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 46(4):441–458MathSciNetCrossRefMATHGoogle Scholar
  12. Cho Y, Sun H, Lee K (2015) Exact likelihood inference for an exponential parameter under generalized progressive hybrid censoring scheme. Stat Methodol 23:18–34MathSciNetCrossRefGoogle Scholar
  13. Cox DR, Oakes D (1985) Analysis of survival data. Monographs on statistics and applied probability, 2nd edn. Chapman and Hall, LondonGoogle Scholar
  14. Cramer E, Balakrishnan N (2013) On some exact distributional results based on Type-I progressively hybrid censored data from exponential distributions. Stat Methodol 10(1):128–150MathSciNetCrossRefMATHGoogle Scholar
  15. Cramer E, Burkschat M, Górny J (2016) On the exact distribution of the MLEs based on Type-II progressively hybrid censored data from exponential distributions. J Stat Comput Simul 86(10):2036–2052MathSciNetCrossRefGoogle Scholar
  16. de Boor C (2001) A practical guide to splines, revised edn. Springer, BerlinMATHGoogle Scholar
  17. Epstein B (1954) Truncated life tests in the exponential case. Ann Math Stat 25(3):555–564MathSciNetCrossRefMATHGoogle Scholar
  18. Ganguly A, Mitra S, Samanta D, Kundu D (2012) Exact inference for the two-parameter exponential distribution under Type-II hybrid censoring. J Stat Plan Inference 142(3):613–625MathSciNetCrossRefMATHGoogle Scholar
  19. Górny J (2017) A new approach to hybrid censoring. Ph.D. thesis, RWTH Aachen University, Aachen, Germany.  https://doi.org/10.18154/RWTH-2017-05192
  20. Górny J, Cramer E (2016) Exact likelihood inference for exponential distributions under generalized progressive hybrid censoring schemes. Stat Methodol 29:70–94MathSciNetCrossRefGoogle Scholar
  21. Górny J, Cramer E (2017) From B-spline representations to gamma representations in hybrid censoring. Stat Pap.  https://doi.org/10.1007/s00362-016-0866-4 Google Scholar
  22. Górny J, Cramer E (2018a) Modularization of hybrid censoring schemes and its application to unified progressive hybrid censoring. Metrika 81(2):173–210MathSciNetCrossRefMATHGoogle Scholar
  23. Górny J, Cramer E (2018b) Type-I hybrid censoring of uniformly distributed lifetimes. Commun Stat Theory Methods.  https://doi.org/10.1080/03610926.2017.1414255 MATHGoogle Scholar
  24. Huang W-T, Yang K-C (2010) A new hybrid censoring scheme and some of its properties. Tamsui Oxford J Math Sci 26(4):355–367MathSciNetMATHGoogle Scholar
  25. Lee K, Sun H, Cho Y (2016) Exact likelihood inference of the exponential parameter under generalized Type II progressive hybrid censoring. J Korean Stat Soc 45(1):123–136MathSciNetCrossRefMATHGoogle Scholar
  26. Nelson WB (2004) Applied life data analysis. Wiley, HobokenMATHGoogle Scholar
  27. Park S, Balakrishnan N (2012) A very flexible hybrid censoring scheme and its Fisher information. J Stat Comput Simul 82(1):41–50MathSciNetCrossRefMATHGoogle Scholar
  28. Schumaker LL (2007) Spline functions: basic theory, 3rd edn. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar

Copyright information

© The Indian Society for Probability and Statistics (ISPS) 2018

Authors and Affiliations

  1. 1.Institute of StatisticsRWTH Aachen UniversityAachenGermany

Personalised recommendations