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Extremes

, Volume 22, Issue 2, pp 343–372 | Cite as

The tail dependograph

  • Cécile MercadierEmail author
  • Olivier Roustant
Article
  • 69 Downloads

Abstract

All characterizations of non-degenerate multivariate tail dependence structures are both functional and infinite-dimensional. Taking advantage of the Hoeffding–Sobol decomposition, we derive new indices to measure and summarize the strength of dependence in a multivariate extreme value analysis. The tail superset importance coefficients provide a pairwise ordering of the asymptotic dependence structure. We then define the tail dependograph, which visually ranks the extremal dependence between the components of the random vector of interest. For the purpose of inference, a rank-based statistic is derived and its asymptotic behavior is stated. These new concepts are illustrated with both theoretical models and real data, showing that our methodology performs well in practice.

Keywords

Global sensitivity analysis Hoeffding–Sobol decomposition Multivariate extreme value analysis Pairwise index Tail dependency graph 

AMS 2000 Subject Classifications

Primary 62J10 62G32 62H20 Secondary 62J15 62G20 62-09 

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Notes

Acknowledgments

The authors would like to thank Christian Genest (McGill University, Montréal, Canada) for fruitful discussions. The first author would also like to thank Roland Denis and Benoît Fabrèges (Institut Camille Jordan, Université de Lyon, France) whose discussions and workshop led to great progress in the codes associated with this paper. The authors would like to thank the editor and referees for their helpful comments.

References

  1. Browne, T., Fort, J.-C., Iooss, B., Le Gratiet, L.: Estimate of quantile-oriented sensitivity indices working paper or preprint (2017)Google Scholar
  2. Cai, J.J., Fougères, A.-L., Mercadier, C.: Environmental data: multivariate extreme value theory in practice. Journal de la Société Française de Statistique 154(2), 178–199 (2013)MathSciNetzbMATHGoogle Scholar
  3. Chastaing, G., Gamboa, F., Prieur, C.: Generalized Hoeffding-Sobol decomposition for dependent variables. Electron. J. Stat. 6, 2420–2448 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. de Haan, L., Ferreira, A.: Extreme value theory. An introduction. Springer series in operations research and financial engineering. Springer, New York (2006)Google Scholar
  5. Efron, B., Stein, C.: The jackknife estimate of variance. Ann. Stat. 9(3), 586–596 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Einmahl, J.H.J., Krajina, A., Segers, J.: An m-estimator for tail dependence in arbitrary dimensions. Ann. Stat. 40(3), 1764–1793 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Einmahl, J.H.J., Kiriliouk, A., Krajina, A., Segers, J.: An M-estimator of spatial tail dependence. J. R. Stat. Soc. Ser. B 78(1), 275–298 (2016)MathSciNetCrossRefGoogle Scholar
  8. Fougères, A.-L.: Multivariate extremes. In: Finkenstädt, B., Rootzén, H. (eds.) Extreme Values in Finance, Telecommunications, and the Environment. Monographs on Stat. and Appl. Prob. 99, Chapter 7, pp. 373-388. Chapman and Hall/CRC (2004)Google Scholar
  9. Fougères, A.-L., de Haan, L., Mercadier, C.: Bias correction in multivariate extremes. Ann. Stat. 43(2), 903–934 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fruth, J., Roustant, O., Kuhnt, S.: Total interaction index: a variance-based sensitivity index for interaction screening. Journal of Statistical Planning and Inference 147, 212–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gumbel, E.J.: Multivariate distributions with given margins and analytical examples. Bulletin de l’Institut International de Statistique 37(3), 363–373 (1960)MathSciNetzbMATHGoogle Scholar
  12. Hoeffding, W.: A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics 19, 293–325 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hofert, M., Hornik, K.: qrmdata: data sets for quantitative risk management practice. R package version 2016-01-03-1 (2016)Google Scholar
  14. Hooker, G.: Discovering additive structure in black box functions. In: Proceedings of KDD 2004, pp. 575–580. ACM DL (2004)Google Scholar
  15. Huang, X.: Statistics of bivariate extremes. PhD Thesis, Erasmus University Rotterdam, Tinbergen Institute Research series No. 22 (1992)Google Scholar
  16. Kereszturi, M., Tawn, J., Jonathan, Ph.: Assessing extremal dependence of north sea storm severity. Ocean Eng. 118, 242–259 (2016)CrossRefGoogle Scholar
  17. Kucherenko, S., Song, S.: Quantile based global sensitivity measures. ArXiv e-prints (2016)Google Scholar
  18. Liu, R., Owen, A.B.: Estimating mean dimensionality of analysis of variance decompositions. J. Am. Stat. Assoc. 101(474), 712–721 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Maume-Deschamps, V., Niang, I.: Estimation of quantile oriented sensitivity indices. Statist. Probab. Lett. 134, 122–127 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Mhalla, L., Chavez-Demoulin, V., Naveau, Ph.: Non-linear models for extremal dependence. J. Multivar. Anal. 159, 49–66 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Muehlenstaedt, T., Roustant, O., Carraro, L., Kuhnt, S.: Data-driven kriging models based on FANOVA-decomposition. Stat. Comput. 22(3), 723–738 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Owen, A., Dick, J., Chen, S.: Higher order sobol’ indices. Information and Inference: A Journal of the IMA 3(1), 59–81 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Peng, Y., Ng, W.: Analysing financial contagion and asymmetric market dependence with volatility indices via copulas. Ann. Finance 8(1), 49–74 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Pujol, G., Iooss, B., Janon, A.: Sensitivity: global sensitivity analysis of model outputs. R package version 1.14.0 (2017)Google Scholar
  25. Ressel, P.: Homogeneous distributions, and a spectral representation of classical mean values and stable tail dependence functions. J. Multivar. Anal. 117, 246–256 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Segers, J.: Max-stable models for multivariate extremes. REVSTAT – Statistical Journal 10(1), 61–82 (2012)MathSciNetzbMATHGoogle Scholar
  27. Smith, R.L.: Max-stable processes and spatial extremes. Dept. of Math., Univ. of Surrey, Guildford GU2 5XH England (1990)Google Scholar
  28. Sobol’, I.M.: Sensitivity estimates for nonlinear mathematical models. Mathematical Modeling and Computational Experiment. Model Algorithm, Code 1(4), 1993 (1995)MathSciNetGoogle Scholar
  29. Stephenson, A.G.: Evd: extreme value distributions. R News 2(2), 31–32 (2002)Google Scholar
  30. Stephenson, A.G.: Simulating multivariate extreme value distributions of logistic type. Extremes 6(1), 49–59 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  31. Sueur, R., Iooss, B., Delage, Th.: Sensitivity analysis using perturbed-law based indices for quantiles and application to an industrial case working paper or preprint (2017)Google Scholar
  32. Tawn, J.A.: Modelling multivariate extreme value distributions. Biometrika 77(2), 245–253 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. Tiago de Oliveira, J.: Structure theory of bivariate extremes, extensions. Estudos de Matematica, Estatistica, e Economicos 7, 165–195 (1962/63)Google Scholar
  34. van der Vaart, A.W.: Asymptotic Statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998)Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille JordanVilleurbanneFrance
  2. 2.Mines Saint-Étienne, Univ. Clermont Auvergne, CNRS, UMR 6158 LIMOS, Institut Henri FayolSaint-ÉtienneFrance

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