Sankhya B

, Volume 80, Issue 2, pp 195–221 | Cite as

Order Restricted Bayesian Analysis of a Simple Step Stress Model

  • Debashis Samanta
  • Debasis KunduEmail author
  • Ayon Ganguly


In this article we consider a simple step stress set up under the cumulative exposure model assumption. At each stress level the lifetime distribution of the experimental units are assumed to follow the generalized exponential distribution. We provide the order restricted Bayesian inference of the model parameters by considering the fact that the expected lifetime of the experimental units are larger in lower stress level. Analysis and the related results are extended to different censoring schemes also. The Bayes estimates and the associated credible intervals of the unknown parameters are constructed using importance sampling technique. We perform extensive simulation experiments both for the complete and censored samples to see the performances of the proposed estimators. We analyze two simulated and one real data sets for illustrative purposes. An optimal value of the stress changing time is obtained by minimizing the total posterior coefficient of variations of the unknown parameters.


Step-stress life-tests Cumulative exposure model Bayes estimate Generalized Exponential distribution Credible interval Censoring scheme Optimality 

AMS (2000) subject classification.

Primary 62N02 Secondary 62F15 62F30 


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The authors would like to thank two unknown reviewers for their valuable comments which have helped us to improve the manuscript significantly.


  1. Abdel-Hamid, A. H. and AL-Hussaini, E. K. (2009). Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with type-I censoring. Computational Statistics and Data Analysis, 53: 1328–1338.Google Scholar
  2. Al-Hussaini, E. K. and Ahsanullah, M. (2015). Exponentiated distributions. AP, Paris, France.Google Scholar
  3. Bagdonavicius, V. B. and Nikulin, M. (2002). Accelerated life models: modeling and statistical analysis. Chapman and Hall CRC Press, Boca Raton, Florida.Google Scholar
  4. Balakrishnan, N. (2009). A synthesis of exact inferential results for exponential step-stress models and associated optimal accelerated life-tests. Metrika, 69: 351–396.Google Scholar
  5. Balakrishnan, N., Beutner, E. and Kateri, M. (2009). Order restricted inference for exponential step-stress models. IEEE Transactions on Reliability, 58: 132–142.Google Scholar
  6. Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. (2003). 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.Google Scholar
  7. Congdon, P. (2006). Bayesian Statistical Modeling. Wiley, New York.Google Scholar
  8. Epstein, B. (1954). Truncated life-test in exponential case. Annals of Mathematical Statistics, 25: 555–564.Google Scholar
  9. Gupta, R. D. and Kundu, D. (1999). Generalized exponential distribution. ustralian and New Zealand Journal of Statistics, 41: 173–188.Google Scholar
  10. Gupta, R. D. and Kundu, D. (2001). Exponentiated exponential distribution: An alternative to gamma and weibull distributions. Biometrical Journal, 43: 117–130.Google Scholar
  11. Kundu, D. and Ganguly, A. (2017). Analysis of step-stress models: existing methods and recent developments. Elsevier/ Academic Press, London, UK.Google Scholar
  12. Nadarajah, S. (2011). The exponentiated exponential distribution. Advance in Statistical Analysis, 95: 219–251.Google Scholar
  13. Nelson, W. B. (1980). Accelerated life testing: step-stress models and data analysis. IEEE Transactions on Reliability, 29: 103–108.Google Scholar
  14. Samanta, D., Ganguly, A., Kundu, D. and Mitra, S. (2017). Order restricted Bayesian inference for exponential simple step-stress model. Communication in Statistics - Simulation and Computation, 46: 1113–1135.Google Scholar
  15. Sediakin, N. M. (1966). On one physical principle in reliability theory. Technical Cybernatics, 3: 80–87.Google Scholar
  16. Zhang, Y. and Meeker, W. Q. (2005). Bayesian life test planning for weibull distribution. Metrika, 61: 237–249.Google Scholar

Copyright information

© Indian Statistical Institute 2017

Authors and Affiliations

  1. 1.Department of StatisticsRabindra MahavidyalayaHooghlyIndia
  2. 2.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKanpurIndia
  3. 3.Department of MathematicsIndian Institute of Technology GuwahatiNorth GuwahatiIndia

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