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Noninformative priors for product of exponential means

  • Kang, Sang Gil (Department of Computer and Data Information, Sangji University) ;
  • Kim, Dal Ho (Department of Statistics, Kyungpook National University) ;
  • Lee, Woo Dong (Department of Data Management, Daegu Haany University)
  • Received : 2015.04.18
  • Accepted : 2015.05.22
  • Published : 2015.05.31

Abstract

In this paper, we develop the noninformative priors for the product of different powers of k means in the exponential distribution. We developed the first and second order matching priors. It turns out that the second order matching prior matches the alternative coverage probabilities, and is the highest posterior density matching prior. Also we revealed that the derived reference prior is the second order matching prior, and Jeffreys' prior and reference prior are the same. We showed that the proposed reference prior matches very well the target coverage probabilities in a frequentist sense through simulation study, and an example based on real data is given.

Keywords

References

  1. Barlow, R. E. and Proschan, F. (1975). Statistical theory of reliability and life testing, Holt, Reinhart and Winston, New York.
  2. 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/10.1080/01621459.1989.10478756
  3. Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors (with discussion). In Bayesian Statistics IV, edited by 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 B, 41, 113-147.
  5. Cox, D. R. and Reid, N. (1987). Orthogonal parameters and approximate conditional inference (with dis- cussion). Journal of Royal Statistical Society B, 49, 1-39.
  6. Datta, G. S. and Ghosh, M. (1995). Some remarks on noninformative priors. Journal of the American Statistical Association, 90, 1357-1363. https://doi.org/10.1080/01621459.1995.10476640
  7. Datta, G. S. and Ghosh, M. (1996). On the invariance of noninformative priors. The Annal of Statistics, 24, 141-159. https://doi.org/10.1214/aos/1033066203
  8. Datta, G. S., Ghosh, M. and Mukerjee, R. (2000). Some new results on probability matching priors. Calcutta Statistical Association Bulletin, 50, 179-192. https://doi.org/10.1177/0008068320000306
  9. Davis, D. J. (1952). An analysis of some failure dfata. Journal of the American Statistical Association, 47, 113-150. https://doi.org/10.1080/01621459.1952.10501160
  10. DiCiccio, T. J. and Stern, S. E. (1994). Frequentist and Bayesian Bartlett correction of test statistics based on adjusted pro le likelihood. Journal of Royal Statistical Society B, 56, 397-408.
  11. Epstein, B. and Sobel, M. (1953). Life testing, Journal of the American Statistical Association, 48, 486-502. https://doi.org/10.1080/01621459.1953.10483488
  12. Ghosh, J. K. and Mukerjee, R. (1992). Noninformative priors (with discussion). In Bayesian Statistics IV, edited by J. M. Bernardo, et al., Oxford University Press, Oxford, 195-210.
  13. Ghosh, J. K. and Mukerjee, R. (1995). Frequentist validity of highest posterior density regions in the presence of nuisance parameters. Statistics & Decisions, 13, 131-139.
  14. Lawless, J. F. (2003). Statistical models and methods for lifetime data, John Wiley and Sons, New York.
  15. Kang, S. G., Kim, D. H. and Lee, W. D. (2013). Noninformative priors for the ratio of parameters of two Maxwell distributions. Journal of the Korean Data & Information Science Society, 24, 643-650. https://doi.org/10.7465/jkdi.2013.24.3.643
  16. Kang, S. G., Kim, D. H. and Lee, W. D. (2014). Noninformative priors for the log-logistic distribution. Journal of the Korean Data & Information Science Society, 25, 227-235. https://doi.org/10.7465/jkdi.2014.25.1.227
  17. Kim, H. J. (2006). On Bayesian estimation of the product of Poisson rates with application to reliability. Communications in Statistics: Simulation and Computation, 35, 47-59. https://doi.org/10.1080/03610910500416181
  18. Kenneth, V. D., John, S. D. and Kenneth, J. S. (1998). Incorporating a geometric mean formula into CPI. Monthly Labor Review, October, 2-7.
  19. 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
  20. Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors. Biometrika, 84, 970-975. https://doi.org/10.1093/biomet/84.4.970
  21. Mukerjee, R. and Reid, N. (1999). On a property of probability matching priors: Matching the alternative coverage probabilities. Biometrika, 86, 333-340. https://doi.org/10.1093/biomet/86.2.333
  22. Raubenheimer, L. (2012). Bayesian inference on nonlinear functions of Poisson rates. South African Statistical Jouranl, 46, 299-326.
  23. Saraccoglul, B., Kinaci, I. and Kundu, D. (2012). On estimation of R = P(Y < X) for exponential distribution under progressive type-II censoring. Journal of Statistical Computation and Simulation, 82, 729-744. https://doi.org/10.1080/00949655.2010.551772
  24. Southwood, T. (1978). Ecological methods with particular reference to the study of insect populations, Chapman & Hall, London.
  25. Stein, C. (1985). On the coverage probability of con dence sets based on a prior distribution. Sequential Methods in Statistics, Banach Center Publications, 16, 485-514. https://doi.org/10.4064/-16-1-485-514
  26. Sun, D. and Ye, K. (1995). Reference prior Bayesian analysis for normal mean products. Journal of the American Statistical Association, 90, 589-597. https://doi.org/10.1080/01621459.1995.10476551
  27. Sun, D. and Ye, K. (1999). Reference priors for a product of normal means when variances are unknown. The Canadian Journal of Statistics, 27, 97-103. https://doi.org/10.2307/3315493
  28. Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika, 76, 604-608. https://doi.org/10.1093/biomet/76.3.604
  29. Welch, B. L. and Peers, H. W. (1963). On formulae for con dence points based on integrals of weighted likelihood. Journal of Royal Statistical Society B, 25, 318-329.
  30. Xia, Z. P., Yu, J. Y., Cheng, L. D., Liu, L. F., and Wang, W. M. (2009). Study on the breaking strength of jute fibers using modified Weibull distribution. Journal of Composites Part A: Applied Science and Manufacturing, 40, 54-59. https://doi.org/10.1016/j.compositesa.2008.10.001
  31. Yfantis, E. A. and Flatman, G. T. (1991). On confidence interval for the product of three means of three normally distributed populations. Journal of Chemometrics, 5, 309-319. https://doi.org/10.1002/cem.1180050317