Advertisement

On Bayesian Inference of \(R=P(Y < X)\) for Weibull Distribution

  • Debasis Kundu
Chapter

Abstract

In this paper, we consider the Bayesian inference on the stress-strength parameter \(R = P(Y < X)\), when X and Y follow independent Weibull distributions. We have considered different cases. It is assumed that the random variables X and Y have different scale parameters and (a) a common shape parameter or (b) different shape parameters. Moreover, both stress and strength may depend on some known covariates also. When the two distributions have a common shape parameter, Bayesian inference on R is obtained based on the assumption that the shape parameter has a log-concave prior, and given the shape parameter, the scale parameters have Dirichlet-Gamma prior. The Bayes estimate cannot be obtained in closed form, and we propose to use Gibbs sampling method to compute the Bayes estimate and also to compute the associated highest posterior density (HPD) credible interval. The results have been extended when the covariates are also present. We further consider the case when the two shape parameters are different. Simulation experiments have been performed to see the effectiveness of the proposed methods. One data set has been analyzed for illustrative purposes and finally, we conclude the paper.

Notes

Acknowledgements

The author would like to thank one unknown referee for his/her many constructive suggestions which have helped to improve the paper significantly.

References

  1. 1.
    Birnbaum, Z.W. 1956. On a use if Mann-Whitney statistics. In Proceedings of third berkeley symposium in mathematical statistics and probability, vol. 1, 13–17. Berkeley, CA: University of California Press.Google Scholar
  2. 2.
    Kotz, S., Y. Lumelskii, and M. Pensky. 2003. The stress-strength model and its generalizations. Singapore: World Scientific Press.CrossRefMATHGoogle Scholar
  3. 3.
    Zhou, W. 2008. Statistical inference for \(P(X < Y)\). Statistics in Medicine 27: 257–279.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kundu, D., and M.Z. Raqab. 2009. Estimation of \(R = P(Y < X)\) for three parameter Weibull distribution. Statistics and Probability Letters 79: 1839–1846.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ventura, L., and W. Racugno. 2011. Recent advances on Bayesian inference for \(P(X < Y)\). Bayesian Analysis 6: 1–18.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kizilaslan, F., and M. Nadar. 2016. Estimation of reliability in a multicomponent stress-strength model based on a bivariate Kumaraswamy distribution. Statistical Papers, to appear.  https://doi.org/10.1007/s00362-016-0765-8.
  7. 7.
    Guttman, I., R.A. Johnson, G.K. Bhattacharya, and B. Reiser. 1988. Confidence limits for stress-strength models with explanatory variables. Technometrics 30: 161–168.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Weerahandi, S., and R.A. Johnson. 1992. Testing reliability in a stress-strength model when \(X\) and \(Y\) are normally distributed. Technometrics 34: 83–91.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guttman, I., and G.D. Papandonatos. 1997. A Bayesian approach to a reliability problem; theory, analysis and interesting numerics. Canadian Journal of Statistics 25: 143–158.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Kundu, D., and R.D. Gupta. 2006. Estimation of \(P(Y < X)\) for Weibull distribution. IEEE Transactions on Reliability Theory 55: 270–280.CrossRefGoogle Scholar
  11. 11.
    Murthy, D.N.P., M. Xie, and R. Jiang. 2004. Weibull Models. New York: Wiley.MATHGoogle Scholar
  12. 12.
    Peña, E.A., and A.K. Gupta. 1990. Bayes estimation for the Marshall-Olkin exponential distribution. Journal of the Royal Statistical Society Series B 52: 379–389.MathSciNetMATHGoogle Scholar
  13. 13.
    Berger, J.O., and D. Sun. 1993. Bayesian analysis for the Poly-Weibull distribution. Journal of the American Statistical Association 88: 1412–1418.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kundu, D., and B. Pradhan. 2011. Bayesian analysis of progressively censored competing risks data. Sankhyā Series B 73: 276–296.Google Scholar
  15. 15.
    Devroye, L. 1984. A simple algorithm for generating random variables using log-concave density function. Computing 33: 247–257.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Chen, M.H., and Q.M. Shao. 1999. Monte carlo estimation of bayesian credible and hpd intervals. Journal of Computational and Graphical Statistics 8: 69–92.MathSciNetGoogle Scholar
  17. 17.
    Kundu, D. 2008. Bayesian inference and life testing plan for the Weibull distribution in presence of progressive censoring. Technometrics 50: 144–154.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Aarset, M.V. 1987. How to identify a bathtub failure rate? IEEE Transactions on Reliability 36: 106–108.Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsIndian Institute of Technology KanpurKanpurIndia

Personalised recommendations