Extremes

, Volume 16, Issue 3, pp 325–350 | Cite as

Bayesian model averaging for multivariate extremes

  • Anne Sabourin
  • Philippe Naveau
  • Anne-Laure Fougères
Article

Abstract

The main framework of multivariate extreme value theory is well-known in terms of probability, but inference and model choice remain an active research field. Theoretically, an angular measure on the positive quadrant of the unit sphere can describe the dependence among very high values, but no parametric form can entirely capture it. The practitioner often makes an assertive choice and arbitrarily fits a specific parametric angular measure on the data. Another statistician could come up with another model and a completely different estimate. This leads to the problem of how to merge the two different fitted angular measures. One natural way around this issue is to weigh them according to the marginal model likelihoods. This strategy, the so-called Bayesian Model Averaging (BMA), has been extensively studied in various context, but (to our knowledge) it has never been adapted to angular measures. The main goal of this article is to determine if the BMA approach can offer an added value when analyzing extreme values.

Keywords

Bayesian model averaging Multivariate extremes Parametric modelling Spectral measure 

AMS 2000 Subject Classifications

62F07 62F15 62H20 62H05 62P12 

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References

  1. Apputhurai, P., Stephenson, A.: Accounting for uncertainty in extremal dependence modeling using bayesian model averaging techniques. J. Stat. Plan. Inference 141(5), 1800–1807 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. Ballani, F., Schlather, M.: A construction principle for multivariate extreme value distributions. Biometrika 98(3) (2011)Google Scholar
  3. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications.Wiley, New York (2004)CrossRefGoogle Scholar
  4. Berk, R.: Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Stat. 37(1), 51–58 (1966)MathSciNetCrossRefMATHGoogle Scholar
  5. Boldi, M.O., Davison, A.C.: A mixture model for multivariate extremes. J. R. Stat. Soc., Ser. B Stat. Methodol. 69(2), 217–229 (2007). doi:10.1111/j.1467-9868.2007.00585.x MathSciNetCrossRefMATHGoogle Scholar
  6. Coles, S., Tawn, J.: Modeling extreme multivariate events. J. R. Stat. Soc. B 53, 377–392 (1991)MathSciNetMATHGoogle Scholar
  7. Cooley, D., Davis, R., Naveau, P.: The pairwise beta distribution: A flexible parametric multivariate model for extremes. J. Multivar. Anal. 101(9), 2103–2117 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. Cowles, M., Carlin, B.: Markov chain monte carlo convergence diagnostics: a comparative review. J. Am. Stat. Assoc., 883–904 (1996)Google Scholar
  9. Einmahl, J., Segers, J.: Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Stat. 37(5B), 2953–2989 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. Einmahl, J., de Haan, L., Piterbarg, V.: Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Stat. 29(5), 1401–1423 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. Geweke, J.: Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: In Bayesian Statistics, pp. 169–193. University Press (1992)Google Scholar
  12. Gneiting, T., Raftery, A.: Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc. 102(477), 359–378 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. Gudendorf, G., Segers, J.: Nonparametric estimation of an extreme-value copula in arbitrary dimensions. J. Multivar. Anal. 102, 37–47 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. Guillotte, S., Perron, F., Segers, J.: Non-parametric bayesian inference on bivariate extremes. J. R. Stat. Soc., Ser. B Stat. Methodol. 73, 377–406 (2011)MathSciNetCrossRefGoogle Scholar
  15. Gumbel, E.: Distributions des valeurs extrˆemes en plusieurs dimensions. Publ. Inst. Stat. Univ. Paris 9, 171–173 (1960)MathSciNetMATHGoogle Scholar
  16. de Haan, L.: Extreme Value Theory, an Introduction, Ferreira, A. Springer Series in Operations Research and Financial Engineering (2006)Google Scholar
  17. Heffernan, J., Tawn, J.: A conditional approach for multivariate extreme values (with discussion). J. R. Stat. Soc., Ser. B Stat. Methodol. 66(3), 497–546 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. Heidelberger, P., Welch, P.D.: A spectral method for confidence interval generation and run length control in simulations. Commun ACM 24, 233–245 (1981). doi:10.1145/358598.358630 MathSciNetCrossRefGoogle Scholar
  19. Hoeting, J., Madigan, D., Raftery, A., Volinsky, C.: Bayesian model averaging: A tutorial. Stat. Sci. 14(4), 382–401 (1999)MathSciNetCrossRefMATHGoogle Scholar
  20. Kass, R., Raftery, A.: Bayes factors. J. Am. Stat. Assoc. 90(430), 773–795 (1995)CrossRefMATHGoogle Scholar
  21. Kass, R., Tierney, L., Kadane, J.: The validity of posterior expansions based on Laplace’s method. Bayesian and Likelihood methods in Statistics and Econometrics 7, 473–488 (1990)Google Scholar
  22. Kleijn, B., van der, V.rt, A.: Misspecification in infinite-dimensional bayesian statistics. Ann. Stat. 34(2), 837–877 (2006)CrossRefMATHGoogle Scholar
  23. Ledford, A., Tawn, J.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996)MathSciNetCrossRefMATHGoogle Scholar
  24. Madigan, D., Raftery, A.: Model selection and accounting for model uncertainty in graphical models using occam’s window. J. Am. Stat. Assoc. 89(428), 1535–1546 (1994)CrossRefMATHGoogle Scholar
  25. Raftery, A., Gneiting, T., Balabdaoui, F., Polakowski, M.: Using bayesian model averaging to calibrate forecast ensembles. Mon. Weather Rev. 133(5), 1155–1174 (2005)CrossRefGoogle Scholar
  26. Ramos, A., Ledford, A.: A. new class of models for bivariate joint tails. J. R. Stat. Soc., Ser. B Stat. Methodol. 71(1), 219–241 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. Resnick, S.: Extreme Values, Regular Variation, and Point Processes, Volume 4 of Applied Probability. A Series of the Applied Probability Trust. Springer-Verlag, New York (1987)Google Scholar
  28. Resnick, S., Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering (2007)Google Scholar
  29. Robert, C.: The Bayesian Choice: from Decision-theoretic Foundations to Computational Implementation. Springer Verlag, New York (2007)Google Scholar
  30. Stephenson, A.: Simulating multivariate extreme value distributions of logistic type. Extremes 6(1), 49–59 (2003)MathSciNetCrossRefMATHGoogle Scholar
  31. Tawn, J.: Modelling multivariate extreme value distributions. Biometrika 77(2), 245 (1990)CrossRefMATHGoogle Scholar
  32. van der Vaart, A.: Asymptotic Statistics (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge University Press, Cambridge, MA (2000)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Anne Sabourin
    • 1
    • 2
  • Philippe Naveau
    • 2
  • Anne-Laure Fougères
    • 1
  1. 1.Université de Lyon, CNRS UMR 5208, Université Lyon 1Institut Camille JordanVilleurbanne cedexFrance
  2. 2.Laboratoire des Sciences du Climat et de l’EnvironnementCNRS-CEA-UVSQGif-sur-YvetteFrance

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