Bayesian Inference for Rayleigh Distribution Under Step-Stress Partially Accelerated Test with Progressive Type-II Censoring with Binomial Removal

  • Manoj KumarEmail author
  • Anurag Pathak
  • Sukriti Soni


In this paper, we propose maximum likelihood estimators (MLEs) and Bayes estimators of parameters of the step-stress partially accelerated life testing of Rayleigh distribution in presence of progressive type-II censoring with binomial removal scheme under Square error loss function, General entropy loss function and Linear exponential loss function . The MLEs and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from Rayleigh distribution. Also, we present to analyze two sets of real data to show its applicability.


Step-stress partially accelerated test MLEs Bayes estimators PT-II CBRs SELF GELF LINEX 



The authors wish to thank editor-in-chief, associate editors and the referee for their valuable comments without which the paper could not have taken its present form.


  1. 1.
    Abd Elfattah AM, Hassan AS, Ziedan DM (2006) Efficiency of maximum likelihood estimators under different censored sampling schemes for Rayleigh distribution. InterstatGoogle Scholar
  2. 2.
    Abdel-Ghani MM (2004) The estimation problem of the log-logistic parameters in step partially accelerated life tests using type-I censored data. Natl Rev Soc Sci 14(2):1–19Google Scholar
  3. 3.
    Bai DS, Chung SW (1992) Optimal design of partially accelerated life tests for the exponential distribution under type-I censoring. IEEE Trans Reliab 41:400–406CrossRefGoogle Scholar
  4. 4.
    Bai DS, Chung SW, Chun YR (1993) The optimal design of PALT for the log-normal distribution under type-I censoring. Reliab Eng Syst Saf 40:85–92CrossRefGoogle Scholar
  5. 5.
    Balakrishnan N, Aggarwala R (2000) Progressive censoring: theory, methods, and applications. Birkhauser, BostonCrossRefGoogle Scholar
  6. 6.
    Balakrishnan N, Sandhu RA (1995) A simple simulational algorithm for generating progressive type-II censored samples. Am Stat 49(2):229–230Google Scholar
  7. 7.
    Balakrishnan N, Cramer E, Kamps U (2001) Bounds for means and variances of progressive type-II censored order statistics. Stat Probab Lett 54:301–315CrossRefGoogle Scholar
  8. 8.
    Bhattacharyya GK, Soejoeti Z (1975) A tampered failure rate model for step-stress accelerated life test. Commun Stat Theory Methods 18(5):627–643Google Scholar
  9. 9.
    Calabria R, Pulcini G (1996) Point estimation under-asymmetric loss functions for life-truncated exponential samples. Commun Stat Theory Methods 25(3):585–600CrossRefGoogle Scholar
  10. 10.
    Cohen AC (1963) Progressively censored samples in life testing. Technometrics 5(3):327–339CrossRefGoogle Scholar
  11. 11.
    Degroot MH, Goel PK (1979) Bayesian estimation and optimal designs in partially accelerated life testing. J Dedic Adv Op Logist Res 26(2):223–235Google Scholar
  12. 12.
    Ghitany ME, Atieh B, Nadarajah S (2008) Lindley distribution and its applications. Math Comput Simul 78:493–506CrossRefGoogle Scholar
  13. 13.
    Goel PK (1971) Some estimation problems in the study of tampered random variables. Department of Statistics, Carnegie Mellon University, PittspurghGoogle Scholar
  14. 14.
    Kamps U, Cramer E (2001) On distributions of generalized order statistics. Statistics 35:269–280CrossRefGoogle Scholar
  15. 15.
    Lalitha S, Mishra A (1996) Modified maxcimum likelihood estimation for Rayleigh distribution. Commun Stat Theory Methods 25:389–401CrossRefGoogle Scholar
  16. 16.
    Metropolis N, Ulam S (1949) The monte carlo method. J Am Stat Assoc 44(247):335–341CrossRefGoogle Scholar
  17. 17.
    Nassar MM, Nada NK (2011) The beta generalized Pareto distribution. J Stat: Adv Theory Appl 6:1–17Google Scholar
  18. 18.
    Nelson W (1990) Accelerated testing: statistical models. Test plans and data analysis. Wiley, NewYorkCrossRefGoogle Scholar
  19. 19.
    Nichols MD, Padgett WJ (2006) A bootstrap control chart for Weibull percentiles. Qual Reliab Eng Int 22:141–151CrossRefGoogle Scholar
  20. 20.
    Rayleigh J (1980) On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Philos Mag 10:73–78CrossRefGoogle Scholar
  21. 21.
    Sen S, Maiti Sudhansu S, Chandra N (2016) The xgamma distribution: statistical properties and application. J Mod Appl Stat 15(1):774–788CrossRefGoogle Scholar
  22. 22.
    Shanker Rao G (2006) Numerical analysis. New Age International (P) Ltd, New DelhiGoogle Scholar
  23. 23.
    Shanker R (2015) Akash distribution and its applications. Int J Probab Stat 4(3):65–75Google Scholar
  24. 24.
    Siddiqui MM (1962) Some problems conectede with Rayleigh distributions. J Res Natl Bur Stand 60D:167–174Google Scholar
  25. 25.
    Singh SK, Singh U, Kumar M (2013) Estimation of parameters of generalized inverted exponential distribution for progressive type-II censored sample with binomial removals. J Probab Stat 1–12Google Scholar
  26. 26.
    Sinha SK, Howlader HA (1993) Credible and HPD intervals of the parameter and reliability of Rayleigh distribution. IEEE Trans Reliab 32:217–220Google Scholar
  27. 27.
    Tahani A, Abushala A, Solimanb Ahmed A (2015) Estimating the Pareto parameters under progressive censoring data for constant-partially accelerated life tests. J Stat Comput Simul 85(5):917–934CrossRefGoogle Scholar
  28. 28.
    Tse SK, Yang C, Yuen HK (2000) Statistical analysis of Weibull distributed life time data under type II progressive censoring with binomial removals. J Appl Stat 27:1033–1043CrossRefGoogle Scholar
  29. 29.
    Varian HR (1985) A Bayesian approach to real estate assessment. PhD thesis, In: Savage LJ, Fienberg SE, Arnold Zellner (eds) Studies in Bayesian econometrics and statistics. North HollandGoogle Scholar
  30. 30.
    Wu SJ, Chang CT (2002) Parameter estimations based on exponential progressive type II censored with binomial removals. Int J Inf Manag Sci 13:37–46Google Scholar
  31. 31.
    Wu SJ, Chang CT (2003) Inference in the Pareto distribution based on progressive type II censoring with random removals. J Appl Stat 30(2):163–172CrossRefGoogle Scholar
  32. 32.
    Xiong C, Ji M (2004) Analysis of grouped and censored data from step-stress life testing. IEEE Trans Reliab 53(1):22–28CrossRefGoogle Scholar
  33. 33.
    Yuen HK, Tse SK (1996) Parameters estimation for Weibull distributed lifetimes under progressive censoring with random removeals. J Stat Comput Simul 55(12):57–71CrossRefGoogle Scholar
  34. 34.
    Zellner A (1986) A Bayesian estimation and prediction using asymmetric loss function. J Am Stat Assoc 81:446–451CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsCentral University of HaryanaMahendergarhIndia
  2. 2.Department of StatisticsCentral University of RajasthanKishangarhIndia

Personalised recommendations