Advertisement

Statistics and Computing

, Volume 22, Issue 2, pp 661–675 | Cite as

A semiparametric Bayesian approach to extreme value estimation

  • Fernando Ferraz do Nascimento
  • Dani Gamerman
  • Hedibert Freitas Lopes
Article

Abstract

This paper is concerned with extreme value density estimation. The generalized Pareto distribution (GPD) beyond a given threshold is combined with a nonparametric estimation approach below the threshold. This semiparametric setup is shown to generalize a few existing approaches and enables density estimation over the complete sample space. Estimation is performed via the Bayesian paradigm, which helps identify model components. Estimation of all model parameters, including the threshold and higher quantiles, and prediction for future observations is provided. Simulation studies suggest a few useful guidelines to evaluate the relevance of the proposed procedures. They also provide empirical evidence about the improvement of the proposed methodology over existing approaches. Models are then applied to environmental data sets. The paper is concluded with a few directions for future work.

Keywords

Bayesian GPD Higher quantiles MCMC Threshold estimation Nonparametric estimation of curves 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asmussen, S.: Applied Probability and Queues. Wiley, New York (1987) zbMATHGoogle Scholar
  2. Behrens, C., Gamerman, D., Lopes, H.F.: Bayesian analysis of extreme events with threshold estimation. Stat. Model. 4, 227–244 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bermudez, P., Turkman, M.A., Turkman, K.F.: A predictive approach to tail probability estimation. Extremes 4, 295–314 (2001) MathSciNetCrossRefGoogle Scholar
  4. Cabras, S., Castellanos, M.A., Gamerman, D.: A default Bayesian approach for regression on extremes. Stat. Model. (2011, accepted) Google Scholar
  5. Castellanos, M.A., Cabras, S.: A default Bayesian procedure for the generalized Pareto distribution. J. Stat. Plan. Inference 137, 473–483 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  6. Coles, S.G.: Extreme Value Theory an Applications. Kluver Academic, Dordrecht (2001) Google Scholar
  7. Coles, S.G., Tawn, J.A.: A Bayesian analysis of extreme rainfall data. Appl. Stat. 45, 463–478 (1996) CrossRefGoogle Scholar
  8. Cunnane, C.: Note on the Poisson assumption in partial duration series model. Water Resour. Res. 15, 489–494 (1979) CrossRefGoogle Scholar
  9. Dalal, S., Hall, W.: Approximating priors by mixtures of natural conjugate priors. J. R. Stat. Soc., Ser. B 45, 278–286 (1983) MathSciNetzbMATHGoogle Scholar
  10. Davison, A.C., Smith, R.L.: Models for exceedances over high thresholds (with discussion). J. R. Stat. Soc., Ser. B 52, 393–342 (1990) MathSciNetzbMATHGoogle Scholar
  11. Dey, D., Kuo, L., Sahu, S.: A Bayesian predictive approach to determining the number of components in a mixture distribution. Stat. Comput. 5, 297–305 (1995) CrossRefGoogle Scholar
  12. Diebolt, J., Robert, C.: Estimation of finite mixture distributions by Bayesian sampling. J. R. Stat. Soc., Ser. B 56, 363–375 (1994) MathSciNetzbMATHGoogle Scholar
  13. Diebolt, J., El-Aroui, M., Garrido, M., Girard, S.: Quasi-conjugate Bayes estimates for gpd parameters and application to heavy tails modelling. Extremes 1, 57–78 (2005) MathSciNetCrossRefGoogle Scholar
  14. Doornik, JA: Ox: Object Oriented Matrix Programming, 4.1 console version. Nuffield College, Oxford University, London (1996) Google Scholar
  15. Embrechts, P., Küppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, New York (1997) zbMATHGoogle Scholar
  16. Fisher, R.A., Tippett, L.H.C.: On the estimation of the frequency distributions of the largest and smallest number of a sample. Proc. Camb. Philos. Soc. 24, 180–190 (1928) zbMATHCrossRefGoogle Scholar
  17. Frigessi, A., Haug, O., Rue, H.: A dynamic mixture model for unsupervised tail estimation without threshold selection. Extremes 5, 219–235 (2002) MathSciNetCrossRefGoogle Scholar
  18. Frühwirth-Schnatter, S.: Markov chain Monte Carlo estimation of classical and dynamic switching and mixture models. J. Am. Stat. Assoc. 96, 194–209 (2001) zbMATHCrossRefGoogle Scholar
  19. Gamerman, D., Lopes, H.F.: Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd ed. Chapman and Hall/CRC, Baton Rouge (2006) zbMATHGoogle Scholar
  20. Gramacy, R., Lee, K.: Bayesian treed Gaussian process models with an application to computer modeling. J. Am. Stat. Assoc. 103, 1119–1130 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  21. Jenkinson, A.F.: The frequency distribution of the annual maximum (or minimum) values of meteorological events. Q. J. R. Meteorol. Soc. 81, 158–171 (1955) CrossRefGoogle Scholar
  22. Lopes, H.F., Nascimento, F.F., Gamerman, D.: Generalized Pareto models with time-varying tail behavior. Technical Report LES:UFRJ, in preparation (2011) Google Scholar
  23. von Mises, R.: La distribution de la plus grande de nvaleurs. Am. Math. Soc. 2, 271–294 (1954) Google Scholar
  24. Nascimento, F.F., Gamerman, D., Lopes, H.F.: Regression models for exceedance data via the full likelihood. Environ. Ecol. Stat. (2011, to appear) Google Scholar
  25. Pickands, J.: Statistical inference using extreme order statistics. Ann. Stat. 3, 119–131 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  26. Richardson, S., Green, P.: On Bayesian analysis of mixtures with an unknown number of components. J. R. Stat. Soc., Ser. B 59, 731–792 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  27. Roberts, G.O., Rosenthal, J.S.: Examples of adaptive mcmc. Journal of Computation and Graphical. Statistics 18, 349–367 (2009) MathSciNetGoogle Scholar
  28. Roeder, K., Wasserman, L.: Practical Bayesian density estimation using mixtures of normals. J. Am. Stat. Assoc. 92, 894–902 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  29. Schwarz, G.: Estimating the dimension of a model. Ann. Stat. 6, 461–464 (1978) zbMATHCrossRefGoogle Scholar
  30. Smith, R.L.: Threshold models for sample extremes. Statistical extremes and applications 621–638 (1984) Google Scholar
  31. Spiegelhalter, D.J., Best, N.G., Carlin, B.P., Linde, A.: Bayesian measures of model complexity and fit. J. R. Stat. Soc. B 64, 583–639 (2002) zbMATHCrossRefGoogle Scholar
  32. Tancredi, A., Anderson, C., O’Hagan, A.: Accounting for threshold uncertainty in extreme value estimation. Extremes 9, 87–106 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  33. Titterington, D., Smith, A.F.M., Makov, U.: Statistical Analysis of Finite Mixture Distributions. Wiley, New York (1985) zbMATHGoogle Scholar
  34. Wiper, M., Rios Insua, D., Ruggeri, F.: Mixtures of gamma distributions with applications. J. Comput. Graph. Stat. 10, 440–454 (2001) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Fernando Ferraz do Nascimento
    • 1
  • Dani Gamerman
    • 2
  • Hedibert Freitas Lopes
    • 3
  1. 1.Departamento de Informática e EstatísticaUniversidade Federal do PiauíTeresinaBrazil
  2. 2.Instituto de MatemáticaFederal University of Rio de JaneiroRio de JaneiroBrazil
  3. 3.Booth School of BusinessThe University of ChicagoChicagoUSA

Personalised recommendations