The tail dependograph


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.

This is a preview of subscription content, log in to check access.


  1. Browne, T., Fort, J.-C., Iooss, B., Le Gratiet, L.: Estimate of quantile-oriented sensitivity indices working paper or preprint (2017)

  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)

    MathSciNet  MATH  Google Scholar 

  3. Chastaing, G., Gamboa, F., Prieur, C.: Generalized Hoeffding-Sobol decomposition for dependent variables. Electron. J. Stat. 6, 2420–2448 (2012)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  Google 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)

  9. Fougères, A.-L., de Haan, L., Mercadier, C.: Bias correction in multivariate extremes. Ann. Stat. 43(2), 903–934 (2015)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  MATH  Google Scholar 

  12. Hoeffding, W.: A class of statistics with asymptotically normal distribution. Annals of Mathematical Statistics 19, 293–325 (1948)

    MathSciNet  Article  MATH  Google Scholar 

  13. Hofert, M., Hornik, K.: qrmdata: data sets for quantitative risk management practice. R package version 2016-01-03-1 (2016)

  14. Hooker, G.: Discovering additive structure in black box functions. In: Proceedings of KDD 2004, pp. 575–580. ACM DL (2004)

  15. Huang, X.: Statistics of bivariate extremes. PhD Thesis, Erasmus University Rotterdam, Tinbergen Institute Research series No. 22 (1992)

  16. Kereszturi, M., Tawn, J., Jonathan, Ph.: Assessing extremal dependence of north sea storm severity. Ocean Eng. 118, 242–259 (2016)

    Article  Google Scholar 

  17. Kucherenko, S., Song, S.: Quantile based global sensitivity measures. ArXiv e-prints (2016)

  18. Liu, R., Owen, A.B.: Estimating mean dimensionality of analysis of variance decompositions. J. Am. Stat. Assoc. 101(474), 712–721 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  19. Maume-Deschamps, V., Niang, I.: Estimation of quantile oriented sensitivity indices. Statist. Probab. Lett. 134, 122–127 (2018)

    MathSciNet  Article  MATH  Google Scholar 

  20. Mhalla, L., Chavez-Demoulin, V., Naveau, Ph.: Non-linear models for extremal dependence. J. Multivar. Anal. 159, 49–66 (2017)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  24. Pujol, G., Iooss, B., Janon, A.: Sensitivity: global sensitivity analysis of model outputs. R package version 1.14.0 (2017)

  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)

    MathSciNet  Article  MATH  Google Scholar 

  26. Segers, J.: Max-stable models for multivariate extremes. REVSTAT – Statistical Journal 10(1), 61–82 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Smith, R.L.: Max-stable processes and spatial extremes. Dept. of Math., Univ. of Surrey, Guildford GU2 5XH England (1990)

  28. Sobol’, I.M.: Sensitivity estimates for nonlinear mathematical models. Mathematical Modeling and Computational Experiment. Model Algorithm, Code 1(4), 1993 (1995)

    MathSciNet  Google 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)

    MathSciNet  Article  MATH  Google 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)

  32. Tawn, J.A.: Modelling multivariate extreme value distributions. Biometrika 77(2), 245–253 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  33. Tiago de Oliveira, J.: Structure theory of bivariate extremes, extensions. Estudos de Matematica, Estatistica, e Economicos 7, 165–195 (1962/63)

  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 

Download references


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.

Author information



Corresponding author

Correspondence to Cécile Mercadier.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mercadier, C., Roustant, O. The tail dependograph. Extremes 22, 343–372 (2019).

Download citation


  • 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