Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
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Noninformative priors for the scale parameter in the generalized Pareto distribution

  • Kang, Sang Gil (Department of Computer and Data Information, Sangji University)
  • Received : 2013.10.01
  • Accepted : 2013.10.30
  • Published : 2013.11.30
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Abstract

In this paper, we develop noninformative priors for the generalized Pareto distribution when the scale parameter is of interest. We developed the rst order and the second order matching priors. We revealed that the second order matching prior does not exist. It turns out that the reference prior and Jeffrey's prior do not satisfy a first order matching criterion, and Jeffreys' prior, the reference prior and the matching prior are different. Some simulation study is performed and a real example is given.

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References

  1. Arnold, B. C. and Press, S. J. (1989). Bayesian estimation and prediction for Pareto data. Journal of the American Statistical Association, 84, 1079-1084. https://doi.org/10.1080/01621459.1989.10478875
  2. Balkema, A. A. and de Haan, L. (1974). Residual lifetime at great age. The Annals of Probability, 2, 792-804. https://doi.org/10.1214/aop/1176996548
  3. Behrens, C., Lopes, H. F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling, 4, 227-244. https://doi.org/10.1191/1471082X04st075oa
  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/10.1080/01621459.1989.10478756
  5. 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.
  6. Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion). Journal of Royal Statistical Society B, 41, 113-147.
  7. Castellanos, M. E. and Cabras, A. (2007). A default Bayesian procedures for the generalized Pareto distribution. Journal of Statistical Planning and Inference, 137, 473-483. https://doi.org/10.1016/j.jspi.2006.01.006
  8. Castillo, E. and Hadi, A. (1997). Fitting the generalized Pareto distribution to data. Journal of the American Statistical Association, 92, 1609-1620. https://doi.org/10.1080/01621459.1997.10473683
  9. Coles, S. G. and Powell, E. A. (1996). Bayesian methods in extreme value modeling: A review and new developments. International Statistical Review, 64, 119-136. https://doi.org/10.2307/1403426
  10. 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
  11. 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
  12. Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds. Journal of Royal Statistical Society B, 52, 393-442.
  13. de Zea Bermudez, P. and Amaral Turkman, M. A. (2003). Bayesian approach to parameter estimation of the generalized Pareto distribution. Test, 12, 259-277. https://doi.org/10.1007/BF02595822
  14. Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997). Modelling extremal events for insurance and finance, Springer, Berlin.
  15. Giles, D. E., Feng, H. and Godwin R. T. (2011). On the bias of the maximum likelihood estimators for the two-parameter Lomax distribution, EconometricsWorking Paper EWP1104, Department of Economics, University of Victoria.
  16. Ghosh, J. K. and Mukerjee, R. (1992). Noninformative priors (with discussion). In Bayesian Statistics IV, edited by J. M. Bernardo, et al., Oxford University, Oxford, 195-210.
  17. Ho, K. (2010). A matching prior for extreme quantile estimation of the generalized Pareto distribution. Journal of Statistical Planning and Inference, 140, 1513-1518. https://doi.org/10.1016/j.jspi.2009.12.012
  18. Hogg, R. V. and Klugman, S. A. (1983). On estimation of long-tailed skewed distributions with actuarial applications. Journal of Econometrics, 23, 91-102. https://doi.org/10.1016/0304-4076(83)90077-5
  19. Hosking, J. R. M. and Wallis, J. R. (1987). Parameter and quantile estimation for the generalized Pareto distribution. Technometrics, 29, 339-349. https://doi.org/10.1080/00401706.1987.10488243
  20. Kang, S. G. (2011). Noninformative priors for the common mean in log-normal distributions. Journal of the Korean Data & Information Science Society, 22, 1241-1250.
  21. Kang, S. G., Kim, D. H. and Lee, W. D. (2012). Noninformative priors for the ratio of the scale parameters in the half logistic distributions. Journal of the Korean Data & Information Science Society, 23, 833-841. https://doi.org/10.7465/jkdi.2012.23.4.833
  22. Kang, S. G., Kim, D. H. and Lee, W. D. (2013). Noninformative priors for the shape parameter in the generalized Pareto distribution. Journal of the Korean Data & Information Science Society, 24, 171-178. https://doi.org/10.7465/jkdi.2013.24.1.171
  23. 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
  24. Mukerjee, R. and Ghosh, M. (1997). Second order probability matching priors. Biometrika, 84, 970-975. Journal of the American Statistical Association, 59, 665-680.
  25. Pickands, J. (1975). Statistical inferences using extreme order statistics. The Annals of Statistics, 3, 119-131. https://doi.org/10.1214/aos/1176343003
  26. Smith, R. L. (1987). Estimating tails of probability distributions. The Annals of Statistics, 15, 1174-1207. https://doi.org/10.1214/aos/1176350499
  27. 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.
  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 confidence points based on integrals of weighted likelihood. Journal of Royal Statistical Society B, 25, 318-329.

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