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Noninformative priors for the shape parameter in the generalized Pareto distribution

  • 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 Asset Management, Daegu Haany University)
  • Received : 2012.10.26
  • Accepted : 2012.12.14
  • Published : 2013.01.31

Abstract

In this paper, we develop noninformative priors for the generalized Pareto distribution when the parameter of interest is the shape parameter. We developed the first 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 satisfies a first order matching criterion, but Jeffrey's prior is not a first order matching prior. Some simulation study is performed and a real example is given.

Keywords

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 modelling: A review and new developments. International Statistical Review, 64, 119-136. https://doi.org/10.2307/1403426
  10. Cox, D. R. and Reid, N. (1987). Parametric orthogonality and approximate conditional inference (with discussion). Journal of Royal Statistical Society B, 49, 1-39.
  11. 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
  12. 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
  13. Davison, A. C. and Smith, R. L. (1990). Models for exceedances over high thresholds. Journal of Royal Statistical Society B, 52, 393-442.
  14. 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
  15. Embrechts, P., Kluppelberg, C. and Mikosch, T. (1997). Modelling extremal events for insurance and finance, Springer, Berlin.
  16. Giles, D. E., Feng, H. and Godwin, R. T. (2011). On the bias of the maximum likelihood estimators for the two-parameter Lomax distribution, Econometrics Working Paper EWP1104, Department of Economics, University of Victoria, Canada.
  17. Ghosh, J. K. and Mukerjee, R. (1992). Noninformative priors (with discussion). In Bayesian Statistics IV, edited by J. M. Bernardo et al., Oxford University, 195-210.
  18. 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
  19. 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
  20. 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
  21. 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.
  22. Kim, D. H., Kang, S. G. and Lee, W. D. (2009). Noninformative priors for Pareto distribution. Journal of the Korean Data & Information Science Society, 20, 1213-1223.
  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. https://doi.org/10.1093/biomet/84.4.970
  25. Pickands, J. (1975). Statistical infernces 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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