Noninformative priors for Pareto distribution

  • 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 : 2009.11.30

Abstract

In this paper, we develop noninformative priors for two parameter Pareto distribution. Specially, we derive Jereys' prior, probability matching prior and reference prior for the parameter of interest. In our case, the probability matching prior is only a first order matching prior and there does not exist a second order matching prior. Some simulation reveals that the matching prior performs better to achieve the coverage probability. A real example is also considered.

Keywords

References

  1. Arnold, B. C. and Press, S. J. (1983). Bayesian inference for Pareto populations. Journal of Econometrics, 21, 287-306. https://doi.org/10.1016/0304-4076(83)90047-7
  2. Arnold, B. C. and Press, S. J. (1989a). Bayesian inference and decision techniques, Ed. P. Goel and A. Zellner, North-Holland, Amsterdam.
  3. Arnold, B. C. and Press, S. J. (1989b). Bayesian estimation and prediction for Pareto data. Journal of the American Statistical Association, 84, 1079-1084. https://doi.org/10.2307/2290086
  4. 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
  5. Berger, J. O. and Bernardo, J. M. (1992). On the development of reference priors (with discussion). Bayesian Statistics IV, Oxford University Press, Oxford.
  6. Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of Royal Statistical Society, Series B, 41, 113-147.
  7. 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.
  8. 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
  9. 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
  10. 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
  11. Geisser, S. (1984). Predicting Pareto and exponential observables. Canadian Journal of Statistics, 12, 143-152. https://doi.org/10.2307/3315178
  12. Geisser, S. (1985). Interval prediction for Pareto and exponential observables. Journal of Econometrics, 29, 173-185 https://doi.org/10.1016/0304-4076(85)90038-7
  13. Kang, S. G., Kim D. H. and Lee, W. D. (2008). Noninformative priors for the common mean of several inverse gaussian populations. Journal of Korean Data and Information Science Society, 19, 401-411.
  14. Lee, J. and Lee, W. D. (2008). Likelihood based inference for the shape parameter of Pareto distribution. Journal of Korean Data and Information Science Society, 19, 1173-1181.
  15. Lwin, T. (1972). Estimation of the tail of the Paretian Law. Scandinavian Actuarial Journal, 55, 170-178.
  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. Nigam, A. M. and Hamdy, H. L. (1987). Bayesian prediction bounds for the Pareto lifetime model. Communications in Statistics, Theory and Methods, 16, 1761-1772. https://doi.org/10.1080/03610928708829470
  19. 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.
  20. Soliman, A. A. (2001). LINEX and quadratic approximate Bayes estimators applied to the Pareto model. Communications in Statistics, Simulation and Computation, 30, 47-62. https://doi.org/10.1081/SAC-100001857
  21. Stein, C. (1985). On the coverage probability of confidence sets based on a prior distribution. Sequential Methods in Statistics, Banach Center Publications, 16, 485-514.
  22. Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika, 76, 604-608. https://doi.org/10.1093/biomet/76.3.604
  23. Tiwari, R. C. Yang, Y. and Zalkikar, J. N. (1996). Bayes estimation for the Pareto failure-model using gibbs sampling. IEEE Trans. Reliability, R-45, 471-476.
  24. Welch, B. L. and Peers, H. W. (1963). On formulae for confidence points based on integrals of weighted likelihood. Journal of Royal Statistical Society, Series B, 25, 318-329.