Estimation of \(P(X>Y)\) for Weibull distribution based on hybrid censored samples

  • A. Asgharzadeh
  • M. Kazemi
  • D. Kundu
Original Article


A hybrid censoring scheme is mixture of Type-I and Type-II censoring schemes. Based on hybrid censored samples, this paper deals with the inference on \(R = P(X>Y)\), when X and Y are two independent Weibull distributions with different scale parameters, but having the same shape parameter. The maximum likelihood estimator (MLE), and the approximate MLE of R are obtained. The asymptotic distribution of the MLE of R is obtained. Based on the asymptotic distribution, the confidence interval of R is constructed. Two bootstrap confidence intervals are also proposed. We consider the Bayesian estimate of R, and propose the corresponding credible interval for R. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a real data set has also been presented for illustrative purposes.


Approximate maximum likelihood estimator Hybrid censoring Maximum likelihood estimator Stress-strength model 


  1. Al-Mutairi DK, Ghitany ME, Kundu D (2013) Inferences on stress-strength reliability from Lindley distributions. Commun Stat Theory Methods 42(8):1443–1463MathSciNetCrossRefzbMATHGoogle Scholar
  2. Asgharzadeh A, Valiollahi R, Raqab MZ (2011) Stress-strength reliability of Weibull distribution based on progressively censored samples. SORT 35:103–124MathSciNetzbMATHGoogle Scholar
  3. Asgharzadeh A, Valiollahi R, Raqab MZ (2013) Estimation of the stressstrength reliability for the generalized logistic distribution. Stat Methodol 15:73–94MathSciNetCrossRefGoogle Scholar
  4. Adimari G, Chiogna M (2006) Partially parametric interval estimation of \(Pr( Y>X)\). Comput Stat Data Anal 51:1875–1891MathSciNetCrossRefzbMATHGoogle Scholar
  5. Awad AM, Azzam MM, Hamdan MA (1981) Some inference results in \(P(Y < X)\) in the bivariate exponential model. Commun Stat Theory Methods 10:2515–2524CrossRefGoogle Scholar
  6. Badar MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites. In: Hayashi T, Kawata K, Umekawa S (eds) Progress in science and engineering composites. ICCM-IV, Tokyo, pp 1129–1136Google Scholar
  7. Baklizi A (2008) Likelihood and Bayesian estimation of \(P(Y < X)\) using lower record values from the generalized exponential distribution. Comput Stat Data Anal 52:3468–3473MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian credible and HPD intervals. J Comput Gr Stat 8:69–92MathSciNetGoogle Scholar
  9. 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. Ann Inst Stat Math 55:319–330MathSciNetzbMATHGoogle Scholar
  10. Ebrahimi N (1990) Estimating the parameter of an exponential distribution from hybrid life test. J Stat Plan Inference 23:255–261MathSciNetzbMATHGoogle Scholar
  11. Efron B (1982) The jackknife, the bootstrap and other re-sampling plans. In: CBMSNSF regional conference series in applied mathematics 34, SIAM, Philadelphia, PAGoogle Scholar
  12. Epstein B (1954) Truncated life tests in the exponential case. Ann Stat 25:555–564MathSciNetCrossRefzbMATHGoogle Scholar
  13. Fairbanks K, Madson R, Dykstra R (1982) A confidence interval for an exponential parameter from a hybrid life test. J Am Stat Assoc 77:137–140MathSciNetCrossRefzbMATHGoogle Scholar
  14. Gilks WR, Richardson S, Spiegelhalter DJ (1995) Markov chain monte carlo in practice. Chapman & Hall, LondonzbMATHGoogle Scholar
  15. Gupta RD, Kundu D (1998) Hybrid censoring schemes with exponential failure distribution. Commun Stat Theory Methods 27:3065–3083CrossRefzbMATHGoogle Scholar
  16. Hall P (1988) Theoretical comparison of bootstrap confidence intervals. Ann Stat 16:927–953MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations. World Scientific, New YorkCrossRefzbMATHGoogle Scholar
  18. Kundu D (2007) On hybrid censored Weibull distribution. J Stat Plan Inference 137:2127–2142MathSciNetCrossRefzbMATHGoogle Scholar
  19. Kundu D, Gupta RD (2005) Estimation of \(P(Y <X)\) for the generalized exponential distribution. Metrika 61:291–308MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kundu D, Gupta RD (2006) Estimation of \(P(Y<X)\) for Weibull distributions. IEEE Trans Reliab 55(2):270–280CrossRefGoogle Scholar
  21. Kundu D, Raqab MZ (2009) Estimation of \(R= P(Y < X)\) for three-parameter Weibull distribution. Stat Probab Lett 79:1839–1846MathSciNetCrossRefzbMATHGoogle Scholar
  22. Nadar M, Kizilaslan F, Papadopoulos A (2014) Classical and Bayesian estimation of \(P(Y < X)\) for Kumaraswamy’s distribution. J Stat Comput Simul 84(7):1505–1529MathSciNetCrossRefzbMATHGoogle Scholar
  23. Raqab MZ, Kundu D (2005) Comparison of different estimators of \(P(Y < X)\) for a scaled Burr type X distribution. Commun Stat Simul Comput 34(2):465–483MathSciNetCrossRefzbMATHGoogle Scholar
  24. Raqab MZ, Madi T, Kundu D (2008) Estimation of \(P(Y < X)\) for the three-parameter generalized exponential distribution. Commun Stat Theory Methods 37:2854–2865MathSciNetCrossRefzbMATHGoogle Scholar
  25. Rezaei S, Tahmasbi R, Mahmoodi M (2010) Estimation of \(P(Y < X)\) for generalized Pareto distribution. J Stat Plan Inference 140:480–494MathSciNetCrossRefzbMATHGoogle Scholar
  26. Saracoglua B, Kinacia I, Kundu D (2012) On estimation of \(R = P(Y < X)\) for exponential distribution under progressive type-II censoring. J Stat Comput Simul 82(5):729–744MathSciNetCrossRefGoogle Scholar
  27. Valiollahi R, Asgharzadeh A, Raqab MZ (2013) Estimation of \(P(Y < X)\) for Weibull distribution under progressive Type-II censoring. Commun Stat Theory Methods 42:4476–4498MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Society for Reliability Engineering, Quality and Operations Management (SREQOM), India and The Division of Operation and Maintenance, Lulea University of Technology, Sweden 2015

Authors and Affiliations

  1. 1.Department of Statistics, Faculty of Mathematical SciencesUniversity of MazandaranBabolsarIran
  2. 2.Department of MathematicsIndian Institute of TechnologyKanpurIndia

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