Statistics and Computing

, Volume 28, Issue 3, pp 525–538 | Cite as

A comparison of dependence function estimators in multivariate extremes



Various nonparametric and parametric estimators of extremal dependence have been proposed in the literature. Nonparametric methods commonly suffer from the curse of dimensionality and have been mostly implemented in extreme-value studies up to three dimensions, whereas parametric models can tackle higher-dimensional settings. In this paper, we assess, through a vast and systematic simulation study, the performance of classical and recently proposed estimators in multivariate settings. In particular, we first investigate the performance of nonparametric methods and then compare them with classical parametric approaches under symmetric and asymmetric dependence structures within the commonly used logistic family. We also explore two different ways to make nonparametric estimators satisfy the necessary dependence function shape constraints, finding a general improvement in estimator performance either (i) by substituting the estimator with its greatest convex minorant, developing a computational tool to implement this method for dimensions \(D\ge 2\) or (ii) by projecting the estimator onto a subspace of dependence functions satisfying such constraints and taking advantage of Bernstein–Bézier polynomials. Implementing the convex minorant method leads to better estimator performance as the dimensionality increases.


Asymmetric logistic model Componentwise maxima Convexity Copula Greatest convex minorant Nonparametric and parametric estimators Pickands dependence function 

Supplementary material

11222_2017_9745_MOESM1_ESM.pdf (40 kb)
Supplementary material 1 (pdf 39 KB) (5 kb)
Supplementary material 2 (zip 4 KB)


  1. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, Hoboken (2004)CrossRefMATHGoogle Scholar
  2. Berghaus, B., Bücher, A., Dette, H.: Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence. J. de la Soc. de Fr. de Stat. 154(1), 116–137 (2013)MathSciNetMATHGoogle Scholar
  3. Bücher, A., Dette, H., Volgushev, S.: New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Stat. 39(4), 1963–2006 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. Camicer, J., Peña, J.: Shape preserving representations and optimality of the Bernstein basis. Adv. Comput. Math. 1, 173–196 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. Capéràa, B.Y.P., Fougères, A.: Estimation of a bivariate extreme value distribution. Extremes 3, 311–329 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. Capéràa, B.Y.P., Fougères, A., Genest, C.: A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84(3), 567–577 (1997)MathSciNetCrossRefMATHGoogle Scholar
  7. Coles, S.G., Tawn, J.A.: Modelling extreme multivariate events. J. R. Stat. Soc. B 53(2), 377–392 (1991)MathSciNetMATHGoogle Scholar
  8. Cooley, D., Naveau, P., Poncet, P.: Variograms for Spatial Max-stable Random Fields, 373–390. Lecture notes in Statistics 187. Springer, New York (2006)Google Scholar
  9. Cormier, E., Genest, C., Nešlehová, J.G.: Using B-splines for nonparametric inference on bivariate extreme-value copulas. Extremes 17, 633–659 (2014)MathSciNetCrossRefMATHGoogle Scholar
  10. Davison, A.C., Gholamrezaee, M.M.: Geostatistics of extremes. Proc. R. Soc. A Math. Phys. Eng. Sci. 468, 581–608 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12(4), 1194–1204 (1984)MathSciNetCrossRefMATHGoogle Scholar
  12. de Haan, L., Resnick, S.I.: Limit theory for multivariate sample extremes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 40, 317–337 (1977)MathSciNetCrossRefMATHGoogle Scholar
  13. Deheuvels, P.: On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Stat. Probab. Lett. 12, 429–439 (1991)MathSciNetCrossRefMATHGoogle Scholar
  14. Einmahl, J.H., de Haan, L., Piterbarg, V.I.: Nonparametric estimation of the spectral measure of an extreme value distribution. Ann. Stat. 29(5), 1401–1423 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. Einmahl, J.H., Segers, J.: Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Stat. 37(5), 2953–2989 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. Fils-Villetard, A., Guillou, A., Segers, J.: Projection estimators of Pickands dependence functions. Can. J. Stat. 36(3), 369–382 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. Fougères, A.: Multivariate extremes. In: Extreme Values in Finance, Telecommunications, and the Environment, chapter pp. 377–388. Chapman and Hall (2004)Google Scholar
  18. Genest, C., Nešlehová, J., Quessy, J.-F.: Tests of symmetry for bivariate copulas. Ann. Inst. Stat. Math. 64, 811–834 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. Genest, C., Segers, J.: Rank-based inference for bivariate extreme-value copulas. Ann. Stat. 37(5B), 2990–3022 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. Gentle, J.E.: Elements of Computational Statistics. Springer, New York (2002)MATHGoogle Scholar
  21. Gudendorf, G., Segers, J.: Extreme-value copulas. In: Jaworski, W.H.P., Durante, F., Rychlik, T. (eds.) Proceedings of the Workshop on Copula Theory and its Applications, pp. 127–146. Springer (2010)Google Scholar
  22. Gudendorf, G., Segers, J.: Nonparametric estimation of an extreme-value copula in arbitrary dimensions. J. Multivar. Anal. 102(1), 37–47 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. Gudendorf, G., Segers, J.: Nonparametric estimation of multivariate extreme-value copulas. J. Stat. Plan. Inference 142(12), 3073–3085 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. Guillotte, S.: A Bayesian estimator for the dependence function of a bivariate extreme-value distribution. Can. J. Stat. 36(3), 383–396 (2008)MathSciNetCrossRefGoogle Scholar
  25. Gumbel, E.J.: Distributions des valeurs extrêmes en plusieurs dimensions. Publ. de l’Inst. de Stat. de l’Univ. Paris 9(294), 171–173 (1960a)MATHGoogle Scholar
  26. Gumbel, E.J.: Bivariate exponential distributions. J. Am.Stat. Assoc. 55, 698–707 (1960b)MathSciNetCrossRefMATHGoogle Scholar
  27. Hall, P., Tajvidi, N.: Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 5, 835–844 (2000)MathSciNetCrossRefMATHGoogle Scholar
  28. Hofert, M., Kojadinovic, I., Mächler, M., Yan, J.: Copula: multivariate dependence with copulas. R package version 0.999-12 (2014)Google Scholar
  29. Huser, R., Davison, A.C., Genton, M.G.: Likelihood estimators for multivariate extremes. Extremes 19, 79–103 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. Jiménez, J.R., Villa-Diharce, E., Flores, M.: Nonparametric estimation of the dependence function in bivariate extreme value distributions. J. Multivar. Anal. 76, 159–191 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. Khoudraji, A.: Contributions à l’étude des copules et à la modélisation de valeurs extrêmes bivariées. Ph.D. thesis. Université Laval, Québec, Canada (1995)Google Scholar
  32. Li, B., Genton, M.G.: Nonparametric identification of copula structures. J. Am. Stat. Assoc. 108(502), 666–675 (2013)MathSciNetCrossRefMATHGoogle Scholar
  33. Liebscher, E.: Construction of asymmetric multivariate copulas. J. Am. Stat. Assoc. 99(10), 2234–2250 (2008)MathSciNetMATHGoogle Scholar
  34. Lorenz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publishing Company, New York (1986)Google Scholar
  35. Marcon, G., Padoan, S.A., Antoniano-Villalobos, I.: Bayesian inference for the extremal dependence. Electron. J. Stat. 10(2), 3310–3337 (2016)MathSciNetCrossRefMATHGoogle Scholar
  36. Marcon, G., Padoan, S.A., Naveau, P., Muliere, P., Segers, J.: Nonparametric estimation of the Pickands dependence function using Bernstein polynomials. J. Stat. Plan. Inference 183, 1–17 (2017)MathSciNetCrossRefMATHGoogle Scholar
  37. Matheron, G.: Suffit-il, pour une covariance, d’être de type positif? Sci. de la Terre, Série Informatique Géologique 26, 51–66 (1987)Google Scholar
  38. Naveau, P., Guillou, A., Cooley, D., Diebolt, J.: Modelling pairwise dependence of maxima in space. Biometrika 96(1), 1–17 (2009)MathSciNetCrossRefMATHGoogle Scholar
  39. Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)MATHGoogle Scholar
  40. Padoan, Sa, Ribatet, M., Sisson, S.A.: Likelihood-based inference for max-stable processes. J. Am. Stat. Assoc. 105(489), 263–277 (2010)MathSciNetCrossRefMATHGoogle Scholar
  41. Pickands, J.: Mutivariate extreme value distributions. Bulletin of the International Statistical Institute (Proceedings of the 43rd Session). pp. 859–878 (1981)Google Scholar
  42. Pickands, J.: Multivariate Negative Exponential and Extreme Value Distributions, vol. 51. Springer, Berlin (1989)MATHGoogle Scholar
  43. Resnick, S.I.: Extreme Values, Regular Variation and Point Processes. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  44. Ressel, P.: Homogeneous distributions–and a spectral representation of classical mean values and stable tail dependence functions. J. Multivar. Anal. 117, 246–256 (2013)MathSciNetCrossRefMATHGoogle Scholar
  45. Sauer, T.: Multivariate Bernstein polynomials and convexity. Comput. Aided Geom. Des. 8(6), 465–478 (1991)MathSciNetCrossRefMATHGoogle Scholar
  46. Schlather, M., Tawn, J.A.: Inequalities for the extremal coefficients of multivariate extreme value distributions. Extremes 5, 87–102 (2002)MathSciNetCrossRefMATHGoogle Scholar
  47. Schlather, M., Tawn, J.A.: A dependence measure for multivariate and spatial extreme values: properties and inference. Biometrika 90(1), 139–156 (2003)MathSciNetCrossRefMATHGoogle Scholar
  48. Segers, J.: Non-Parametric Inference for Bivariate Extreme-Value Copulas. In: Ahsanullah, M., Kirmani, S.N.U.A. (eds.) Topics in Extreme Values, pp. 181–203. Nova Science Publishers Inc, New York (2007)Google Scholar
  49. Smith, R.L.: Max-stable processes and spatial extremes. Unpublished manuscript. University of North Carolina, Chapel Hill, U. S. (1990)Google Scholar
  50. Smith, R.L., Tawn, J.A., Yuen, H.K.: Statistics of multivariate extremes. Int. Stat. Rev. 58(1), 47–58 (1990)CrossRefMATHGoogle Scholar
  51. Stephenson, A.G.: High-dimensional parametric modelling of multivariate extreme events. Aust. N. Z. J. Stat. 51(1), 77–88 (2009)MathSciNetCrossRefMATHGoogle Scholar
  52. Tawn, J.A.: Bivariate extreme value theory: models and estimation. Biometrika 75(3), 397–415 (1988)MathSciNetCrossRefMATHGoogle Scholar
  53. Tawn, J.A.: Modelling multivariate extreme value distributions. Biometrika 77(2), 245–253 (1990)MathSciNetCrossRefMATHGoogle Scholar
  54. Varin, B.C., Vidoni, P.: A note on composite likelihood inference and model selection. Biometrika 92(3), 519–528 (2005)MathSciNetCrossRefMATHGoogle Scholar
  55. Varin, C.: On composite marginal likelihoods. AStA Adv. Stat. Anal. 92(1), 1–28 (2008)MathSciNetCrossRefMATHGoogle Scholar
  56. Zhang, D., Wells, M.T., Peng, L.: Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Multivar. Anal. 99(4), 577–588 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Sabrina Vettori
    • 1
  • Raphaël Huser
    • 1
  • Marc G. Genton
    • 1
  1. 1.CEMSE DivisionKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

Personalised recommendations