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 In this paper, a step-stress accelerated life test with two stress variables for Weibull distribution under progressive type-I censoring is considered. The stress-life relationship as a log-linear function of stress levels, and for each combination of stress levels, a cumulative exposure model is assumed. The maximum likelihood and Bayes estimates of the model parameters are obtained. The optimum test plan is developed using variance-optimality criterion, which consists in finding out the optimal stress change time by minimizing asymptotic variance of the maximum likelihood estimates of the log of the scale parameter at the design stress. The proposed study illustrated by using simulated data.


Optimum test plan Step-stress test Weibull distribution Progressive censoring Markov Chain Monte Carlo.

Article Details

Author Biographies

Mashroor Ahmad Khan, Department of Statistics, Pondicherry University, India

Department of Statistics

Navin Chandra, Department of Statistics, Pondicherry University, India

Department of Statistics

How to Cite
Khan, M. A., & Chandra, N. (2021). Optimal Plan and Estimation for Bivariate Step-Stress Accelerated Life Test under Progressive Type-I Censoring. Pakistan Journal of Statistics and Operation Research, 17(3), 683-694.


  1. Bai, D., Kim, M., and Lee, S. (1989). Optimum simple step-stress accelerated life tests with censoring. IEEE transactions on reliability, 38(5):528–532. DOI:
  2. Balakrishnan, N. (2007). Progressive censoring methodology: an appraisal. Test, 16(2):211. DOI:
  3. Balakrishnan, N. and Aggarwala, R. (2000). Progressive censoring: theory, methods, and applications. Springer Science and Business Media. DOI:
  4. Balakrishnan, N. and Han, D. (2009). Optimal step-stress testing for progressively type-i censored data from exponential distribution. Journal of statistical planning and inference, 139(5):1782–1798. DOI:
  5. Chandra, N. and Khan, M. A. (2015). Optimum generalized compound linear plan for multiple-step step-stress accelerated life tests. International Journal of Mathematical Modelling and Computations, 5(3):267–275.
  6. Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2003). Bayesian Data Analysis. Chapman and Hall/CRC, New York. DOI:
  7. Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical science, 7(4):457–472. DOI:
  8. Gouno, E., Sen, A., and Balakrishnan, N. (2004). Optimal step-stress test under progressive type i censoring. IEEE transactions on reliability, 53(3):383–393. DOI:
  9. Hakamipour, H. and Rezaei, S. (2015). Optimal design for a bivariate simple step-stress acceleratedlife testing model with type-ii censoring and gompertz distribution. International Journal of Information Technology and Decision Making, 14(6):1243–1262. DOI:
  10. Khamis, I. H. (1997). Optimum m-step step-stress design with k stress variables. Communications in Statis- tics - Simulation and Computation, 26(4):1301–1313. DOI:
  11. Khamis, I. H. and Higgins, J. J. (1996). Optimum 3-step step-stress tests. IEEE transactions on reliability, 45(2):341–345. DOI:
  12. Khamis, I. H. and Higgins, J. J. (1998). A new model for step-stress testing. IEEE transactions on reliability, 47(2):131–134. DOI:
  13. Lee, J. and Pan, R. (2008). Bayesian inference model for step-stress accelerated life testing with type-ii censoring. In 2008 Annual Reliability and Maintainability Symposium, pages 91–96.
  14. Li, C. and Fard, N. (2007). Optimum bivariate step-stress accelerated life test for censored data. IEEE Transactions on Reliability, 56(1):77–84. DOI:
  15. Ling, L., Xu, W., and Li, M. (2011). Optimal bivariate step-stress accelerated life test for type-i hybrid censored data. Journal of Statistical Computation and Simulation, 81(9):1175–1186. DOI:
  16. Minford, W. (1982). Accelerated life testing and reliability of high k multilayer ceramic capacitors. IEEE Transactions on Components, Hybrids, and Manufacturing Technology, 5(3):297–300. DOI:
  17. Mogilevski, B. M. and Shirn, G. A. (1988). Accelerated life tests of ceramic capacitors. In Electronics Components Conference, pages 362–370. DOI:
  18. Munikoti, R. and Dhar, P. (1988). Highly accelerated life testing (halt) for multilayer ceramic capacitor qualification. IEEE Transactions on Components, Hybrids, and Manufacturing Technology, 11(4):342–345. DOI:
  19. Nelson, W. (1980). Accelerated life testing step-stress models and data analysis. IEEE Transactions on Reliability, R-29(2):103–108. DOI:
  20. Sha, N. and Pan, R. (2014). Bayesian analysis for step-stress accelerated life testing using weibull propor- tional hazard model. Statistical Papers, 55(3):715–726. DOI:
  21. Sinha, S. K. (1998). Bayesian estimation.
  22. Spiegelhalter, D., Thomas, A., Best, N., and Lunn, D. (2003). WinBUGS user manual.
  23. Van Dorp, J. R. and Mazzuchi, T. A. (2004). A general bayes exponential inference model for accelerated life testing. Journal of statistical planning and inference, 119(1):55–74. DOI:
  24. Van Dorp, J. R., Mazzuchi, T. A., Fornell, G. E., and Pollock, L. R. (1996). A bayes approach to step-stress accelerated life testing. IEEE Transactions on Reliability, 45(3):491–498. DOI: