An objective Bayesian analysis for multiple step stress accelerated life tests

  • Kim, Dal-Ho (Department of Statistics, Kyungpook National University) ;
  • Kang, Sang-Gil (Department of Applied Statistics, Sangji University) ;
  • Lee, Woo-Dong (Department of Asset Management, Daegu Haany University)
  • Published : 2009.05.31

Abstract

This paper derives noninformative priors for scale parameter of exponential distribution when the data are collected in multiple step stress accelerated life tests. We nd the objective priors for this model and show that the reference prior satisfies first order matching criterion. Also, we show that there exists no second order matching prior. Some simulation results are given and using artificial data, we perform Bayesian analysis for proposed priors.

References

  1. Bhattacharyya, G. K. and Soejoeti, Z. (1989). A tampered failure rate model for step stress accelerated life test. Communications in Statistics Theory and Methods, 18, 1627-1643. https://doi.org/10.1080/03610928908829990
  2. Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of peans : Bayesian analysis with reference priors. Journal of the American Statistical Association, 84, 200-207. https://doi.org/10.2307/2289864
  3. Berger, J. O. and Bernardo, J. M. (1992a). On the development of reference priors (with discussion). Bayesian Statistics IV, J. M. Bernardo, et. al., Oxford University Press, Oxford, 35-60.
  4. Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of Royal Statistical Society, Series B, 41, 113-147.
  5. Cox, D. R. and Reid, N. (1987). Orthogonal parameters and approximate conditional inference (with discussion). Journal of Royal Statistical Society, Series B, 49, 1-39.
  6. Datta, G. S. and Ghosh, J. K. (1995a). On priors providing frequentist validity for Bayesian inference. Biometrika, 82, 37-45. https://doi.org/10.1093/biomet/82.1.37
  7. Datta, G. S. and Ghosh, M. (1995b). Some remarks on noninformative priors. Journal of the American Statistical Association, 90, 1357-1363. https://doi.org/10.2307/2291526
  8. Datta, G. S. and Ghosh, M. (1996). On the invariance of noninformative priors. The Annals of Statistics, 24, 141-159. https://doi.org/10.1214/aos/1033066203
  9. DeGroot and Goel (1979). Bayesian estimation and optimal design in partially accelerated life testing. Naval Research Logistics Quarterly, 26, 223-235. https://doi.org/10.1002/nav.3800260204
  10. Ghosh, M. (1992). On some Bayesian solutions of the Neyman-Scott problem. Technical Report Number 407, Department of Statistics, University of Florida.
  11. Ghosh, J. K. and Mukerjee, R. (1992). Noninformative priors (with discussion). Bayesian Statistics IV, J. M. Bernardo, et. al., Oxford University Press, Oxford, 195-210.
  12. Jeffreys, H. (1961). Theory of probability, Oxford University Press, New York.
  13. Kang S., Kim D. and Lee W. (2008a). Noninformative priors for the common mean of several inverse Gaussian populations. Journal of the Korean Data and Information Science Society, 19, 401-411.
  14. Kang S., Kim D. and Lee W. (2008b). Reference priors for the location parameter in the exponential distributions. Journal of the Korean Data and Information Science Society, 19, 1409-1418.
  15. Madi, M. T. (1993). Multiple step stress accelerated life test: The tampered failure rate model. Communications in Statistics, Theory and Methods, 22, 2631-2639.
  16. Mukerjee, R. and Dey, D. K. (1993). Frequentist validity of posterior quantiles in the presence of a nuisance parameter: Higher order asymptotics. Biometrika, 80, 499-505. https://doi.org/10.1093/biomet/80.3.499
  17. Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors. Biometrika, 84, 970-975. https://doi.org/10.1093/biomet/84.4.970
  18. Nelson, W. (1980). Accelerated life testing-step stress models and data analysis. IEEE Transactions on Reliability, R-29, 103-108. https://doi.org/10.1109/TR.1980.5220742
  19. Pathak, P. K. Singh, A. K. and Zimmer, W. J. (1987). Empirical Bayesian estimation of mean life from an accelerated life test. Journal of Statistical Planning and Inference, 35, 353-363.
  20. Peers, H. W. (1965). On confidence sets and Bayesian probability points in the case of several parameters. Journal of Royal Statistical Society, Series B, 27, 9-16.
  21. Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics, 5, 375-383. https://doi.org/10.2307/1266340
  22. Shaked, M. and Singpurwalla, N. D. (1983). Inference for step stress accelerated life tests. Journal of Statistical Planning and Inference, 7, 295-306. https://doi.org/10.1016/0378-3758(83)90001-0
  23. Stein, C. (1985). On the coverage probability of confidence sets based on a prior distribution. Sequential Methods in Statistics, 16, 485-514.
  24. Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika, 76, 604-608. https://doi.org/10.1093/biomet/76.3.604
  25. Welch, B. N. and Peers, B. (1963). On formulae for confidence points based on integrals of weighted likelihood. Journal of Royal Statistical Society, Series B, 35, 318-329.