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.
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References
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)
Ballani, F., Schlather, M.: A construction principle for multivariate extreme value distributions. Biometrika 98(3) (2011)
Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications.Wiley, New York (2004)
Berk, R.: Limiting behavior of posterior distributions when the model is incorrect. Ann. Math. Stat. 37(1), 51–58 (1966)
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
Coles, S., Tawn, J.: Modeling extreme multivariate events. J. R. Stat. Soc. B 53, 377–392 (1991)
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)
Cowles, M., Carlin, B.: Markov chain monte carlo convergence diagnostics: a comparative review. J. Am. Stat. Assoc., 883–904 (1996)
Einmahl, J., Segers, J.: Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Stat. 37(5B), 2953–2989 (2009)
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)
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)
Gneiting, T., Raftery, A.: Strictly proper scoring rules, prediction, and estimation. J. Am. Stat. Assoc. 102(477), 359–378 (2007)
Gudendorf, G., Segers, J.: Nonparametric estimation of an extreme-value copula in arbitrary dimensions. J. Multivar. Anal. 102, 37–47 (2011)
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)
Gumbel, E.: Distributions des valeurs extrˆemes en plusieurs dimensions. Publ. Inst. Stat. Univ. Paris 9, 171–173 (1960)
de Haan, L.: Extreme Value Theory, an Introduction, Ferreira, A. Springer Series in Operations Research and Financial Engineering (2006)
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)
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
Hoeting, J., Madigan, D., Raftery, A., Volinsky, C.: Bayesian model averaging: A tutorial. Stat. Sci. 14(4), 382–401 (1999)
Kass, R., Raftery, A.: Bayes factors. J. Am. Stat. Assoc. 90(430), 773–795 (1995)
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)
Kleijn, B., van der, V.rt, A.: Misspecification in infinite-dimensional bayesian statistics. Ann. Stat. 34(2), 837–877 (2006)
Ledford, A., Tawn, J.: Statistics for near independence in multivariate extreme values. Biometrika 83(1), 169–187 (1996)
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)
Raftery, A., Gneiting, T., Balabdaoui, F., Polakowski, M.: Using bayesian model averaging to calibrate forecast ensembles. Mon. Weather Rev. 133(5), 1155–1174 (2005)
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)
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)
Resnick, S., Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering (2007)
Robert, C.: The Bayesian Choice: from Decision-theoretic Foundations to Computational Implementation. Springer Verlag, New York (2007)
Stephenson, A.: Simulating multivariate extreme value distributions of logistic type. Extremes 6(1), 49–59 (2003)
Tawn, J.: Modelling multivariate extreme value distributions. Biometrika 77(2), 245 (1990)
van der Vaart, A.: Asymptotic Statistics (Cambridge Series in Statistical and Probabilistic Mathematics). Cambridge University Press, Cambridge, MA (2000)
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Sabourin, A., Naveau, P. & Fougères, AL. Bayesian model averaging for multivariate extremes. Extremes 16, 325–350 (2013). https://doi.org/10.1007/s10687-012-0163-0
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DOI: https://doi.org/10.1007/s10687-012-0163-0