Reference priors for two parameter exponential stress-strength model

  • Kang, Sang-Gil (Department of Data Information, Sangji University) ;
  • Kim, Dal-Ho (Department of Statistics, Kyungpook National University) ;
  • Le, Woo-Dong (Department of Asset Management, Daegu Haany University)
  • Received : 2010.06.29
  • Accepted : 2010.09.15
  • Published : 2010.09.30

Abstract

In this paper, we develop the noninformative priors for the reliability in a stress-strength model where a strength X and a stress Y have independent exponential distributions with different scale parameters and a common location parameter. We derive the reference priors and prove the propriety of joint posterior distribution under the general prior including the reference priors. Through the simulation study, we show that the proposed reference priors match the target coverage probabilities in a frequentist sense.

Keywords

References

  1. Baklizi, A. and El-Masri, E. Q. (2004). Shrinkage estimation of P(X < Y) in the exponential case with common location parameter. Metrika, 59, 163-171. https://doi.org/10.1007/s001840300277
  2. Bai, D. S. and Hong, Y. W. (1992). Estimation of P(X < Y) in the exponential case with common location parameter. Communications in Statistics-Theory and Methods, 21, 269-282. https://doi.org/10.1080/03610929208830777
  3. Barlow, R. E. and Proschan, F. (1975). Statistical theory of reliability and life testing, Holt, Reinhart and Winston, New York.
  4. Beg M. A. (1980). Estimation of P(X < Y) for truncation parameter distribution. Communications in Statistics-Theory and Methods, 9, 327-345. https://doi.org/10.1080/03610928008827882
  5. Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors. Journal of the American Statistical Association, 84, 200-207. https://doi.org/10.2307/2289864
  6. Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors (with discussion). Bayesian Statistics IV, J.M. Bernardo et al., Oxford University Press, Oxford, 35-60.
  7. Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of Royal Statistical Society, B, 41, 113-147.
  8. Datta, G. S. (1996). On priors providing frequentist validity for Bayesian inference for multiple parametric functions. Biometrika, 83, 287-298. https://doi.org/10.1093/biomet/83.2.287
  9. Datta, G. S. and Ghosh, J. K. (1995). On priors providing frequentist validity for Bayesian inference. Biometrika, 82, 37-45. https://doi.org/10.1093/biomet/82.1.37
  10. Davis, D. J. (1952). An analysis of some failure data. Journal of the American Statistical Association, 47, 113-150. https://doi.org/10.2307/2280740
  11. DiCiccio, T. J. and Stern, S. E. (1994). Frequentist and Bayesian Bartlett correction of test statistics based on adjusted profile likelihood. Journal of Royal Statistical Society, B, 56, 397-408.
  12. Epstein, B. and Sobel, M. (1953). Life testing. Journal of the American Statistical Association, 48, 486-502. https://doi.org/10.2307/2281004
  13. Ghosal, S. (1997). Reference priors in multiparameter nonregular cases. Test, 6, 159-186. https://doi.org/10.1007/BF02564432
  14. Ghosal, S. (1999). Probability matching priors for non-regular cases. Biometrika, 86, 956-964. https://doi.org/10.1093/biomet/86.4.956
  15. Ghosal, S. and Samanta, T. (1995). Asymptotic behavior of Bayes estimates and posterior distributions in multiparameter nonregular cases. Mathematical Methods of Statistics, 4, 361-388.
  16. Ghosal, S. and Samanta, T. (1997). Expansion of Bayes risk for entropy loss and reference prior in nonregular cases. Statistics and Decisions, 15, 129-140.
  17. 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.
  18. Kang, S. G., Kim, D. H. and Lee, W. D. (2008). Reference priors for the location parameter in the exponential distributions. Journal of the Korean Data & Information Science Society, 19, 1409-1418.
  19. Kang, S. G., Kim, D. H. and Lee, W. D. (2010). Reference priors for the common location parameter in half-normal distributions. Journal of the Korean Data & Information Science Society, Accepted.
  20. Kim, D. H., Kang, S. G. and Lee, W. D. (2009a). An objective Bayesian analysis for multiple step stress accelerated life tests. Journal of the Korean Data & Information Science Society, 20, 601-614.
  21. Kim, D. H., Kang, S. G. and Lee, W. D. (2009b). Noninformative priors for Pareto distribution. Journal of the Korean Data & Information Science Society, 20, 1213-1223.
  22. Krishnamoorthy, K., Mukherjee, S. and Guo, H. (2007). Inference on reliability in two-parameter exponential stress-strength model. Metrika, 65, 261-273. https://doi.org/10.1007/s00184-006-0074-7
  23. Lawless, J. F. (2003). Statistical models and methods for lifetime data, Wiley, New York.
  24. Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors. Biometrika, 84, 970-975. https://doi.org/10.1093/biomet/84.4.970
  25. Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika, 76, 604-608. https://doi.org/10.1093/biomet/76.3.604
  26. Welch, B. L. and Peers, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihood. Journal of Royal Statistical Society, B, 25, 318-329.