• Bernoulli
  • Volume 22, Number 1 (2016), 213-241.

Polynomial Pickands functions

Simon Guillotte and François Perron

Full-text: Open access


Pickands dependence functions characterize bivariate extreme value copulas. In this paper, we study the class of polynomial Pickands functions. We provide a solution for the characterization of such polynomials of degree at most $m+2$, $m\geq0$, and show that these can be parameterized by a vector in $\mathbb{R}^{m+1}$ belonging to the intersection of two ellipsoids. We also study the class of Bernstein approximations of order $m+2$ of Pickands functions which are shown to be (polynomial) Pickands functions and parameterized by a vector in $\mathbb{R}^{m+1}$ belonging to a polytope. We give necessary and sufficient conditions for which a polynomial Pickands function is in fact a Bernstein approximation of some Pickands function. Approximation results of Pickands dependence functions by polynomials are given. Finally, inferential methodology is discussed and comparisons based on simulated data are provided.

Article information

Bernoulli, Volume 22, Number 1 (2016), 213-241.

Received: May 2013
Revised: June 2014
First available in Project Euclid: 30 September 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bernstein’s theorem extreme value copulas Lorentz degree Pickands dependence function polynomials spectral measure


Guillotte, Simon; Perron, François. Polynomial Pickands functions. Bernoulli 22 (2016), no. 1, 213--241. doi:10.3150/14-BEJ656.

Export citation


  • [1] Beirlant, J., Goegebeur, Y., Teugels, J. and Segers, J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. Chichester: Wiley. With contributions from Daniel De Waal and Chris Ferro.
  • [2] Boldi, M.-O. and Davison, A.C. (2007). A mixture model for multivariate extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 217–229.
  • [3] Bücher, A., Dette, H. and Volgushev, S. (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39 1963–2006.
  • [4] Capéraà, P., Fougères, A.-L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567–577.
  • [5] Coles, S.G. and Tawn, J.A. (1991). Modelling extreme multivariate events. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 377–392.
  • [6] Coles, S.G. and Tawn, J.A. (1994). Statistical methods for multivariate extremes: An application to structural design. Appl. Statist. 43 1–48.
  • [7] de Haan, L., Neves, C. and Peng, L. (2008). Parametric tail copula estimation and model testing. J. Multivariate Anal. 99 1260–1275.
  • [8] Dupuis, D.J. and Tawn, J.A. (2001). Effects of mis-specification in bivariate extreme value problems. Extremes 4 315–330 (2002).
  • [9] Einmahl, J.H.J., Krajina, A. and Segers, J. (2008). A method of moments estimator of tail dependence. Bernoulli 14 1003–1026.
  • [10] Fils-Villetard, A., Guillou, A. and Segers, J. (2008). Projection estimators of Pickands dependence functions. Canad. J. Statist. 36 369–382.
  • [11] Fougères, A.-L., Mercadier, C. and Nolan, J.P. (2013). Dense classes of multivariate extreme value distributions. J. Multivariate Anal. 116 109–129.
  • [12] Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990–3022.
  • [13] Ghoudi, K., Khoudraji, A. and Rivest, L.-P. (1998). Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canad. J. Statist. 26 187–197.
  • [14] Guillotte, S., Perron, F. and Segers, J. (2011). Non-parametric Bayesian inference on bivariate extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 377–406.
  • [15] Hall, P. and Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 6 835–844.
  • [16] Hudson, H.M. (1978). A natural identity for exponential families with applications in multiparameter estimation. Ann. Statist. 6 473–484.
  • [17] Joe, H., Smith, R.L. and Weissman, I. (1992). Bivariate threshold methods for extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 171–183.
  • [18] Johnson, N.L. (1957). A note on the mean deviation of the binomial distribution. Biometrika 44 532–533.
  • [19] Karlin, S. and Shapley, L.S. (1953). Geometry of moment spaces. Mem. Amer. Math. Soc. 1953 93.
  • [20] Klüppelberg, C. and May, A. (2006). Bivariate extreme value distributions based on polynomial dependence functions. Math. Methods Appl. Sci. 29 1467–1480.
  • [21] Ledford, A.W. and Tawn, J.A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83 169–187.
  • [22] Pólya, G. and Szegő, G. (1998). Problems and Theorems in Analysis. II. Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry. Classics in Mathematics. Berlin: Springer. Translated from the German by C.E. Billigheimer. Reprint of the 1976 English translation.
  • [23] Powers, V. and Reznick, B. (2000). Polynomials that are positive on an interval. Trans. Amer. Math. Soc. 352 4677–4692.
  • [24] Roberts, A.W. and Varberg, D.E. (1973). Convex Functions. Pure and Applied Mathematics 57. New York–London: Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers].
  • [25] Sancetta, A. and Satchell, S. (2001). Bernstein approximation to copula function and portfolio optimization. DAE working paper, Univ. Cambridge.
  • [26] Sancetta, A. and Satchell, S. (2004). The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20 535–562.
  • [27] Segers, J. (2007). Nonparametric inference for bivariate extreme-value copulas. In Topics in Extreme Values (M. Ahsanullah and S.N.U.A. Kirmani, eds.) 181–203. New York: Nova Science Publishers.
  • [28] Smith, R.L. (1994). Multivariate threshold methods. In Extreme Value Theory and Applications (J. Galambos, J. Lechner and E. Simiu, eds.) 225–248. Dordrecht: Kluwer.
  • [29] Szegő, G. (1975). Orthogonal Polynomials, 4th ed. Colloquium Publications XXIII. Providence, RI: Amer. Math. Soc.
  • [30] Tawn, J.A. (1988). Bivariate extreme value theory: Models and estimation. Biometrika 75 397–415.
  • [31] Weissman, I. (2008). On some dependence measures for multivariate extreme value distributions. In Advances in Mathematical and Statistical Modeling. Stat. Ind. Technol. 171–180. Boston, MA: Birkhäuser.