• Bernoulli
  • Volume 14, Number 4 (2008), 1003-1026.

A method of moments estimator of tail dependence

John H.J. Einmahl, Andrea Krajina, and Johan Segers

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In the world of multivariate extremes, estimation of the dependence structure still presents a challenge and an interesting problem. A procedure for the bivariate case is presented that opens the road to a similar way of handling the problem in a truly multivariate setting. We consider a semi-parametric model in which the stable tail dependence function is parametrically modeled. Given a random sample from a bivariate distribution function, the problem is to estimate the unknown parameter. A method of moments estimator is proposed where a certain integral of a nonparametric, rank-based estimator of the stable tail dependence function is matched with the corresponding parametric version. Under very weak conditions, the estimator is shown to be consistent and asymptotically normal. Moreover, a comparison between the parametric and nonparametric estimators leads to a goodness-of-fit test for the semiparametric model. The performance of the estimator is illustrated for a discrete spectral measure that arises in a factor-type model and for which likelihood-based methods break down. A second example is that of a family of stable tail dependence functions of certain meta-elliptical distributions.

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Bernoulli Volume 14, Number 4 (2008), 1003-1026.

First available in Project Euclid: 6 November 2008

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asymptotic properties confidence regions goodness-of-fit test meta-elliptical distribution method of moments multivariate extremes tail dependence


Einmahl, John H.J.; Krajina, Andrea; Segers, Johan. A method of moments estimator of tail dependence. Bernoulli 14 (2008), no. 4, 1003--1026. doi:10.3150/08-BEJ130.

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  • [1] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004)., Statistics of Extremes: Theory and Applications. Chichester: Wiley.
  • [2] Bingham, N.C., Goldie, C.M. and Teugels, J.L. (1987)., Regular Variation. Cambridge: Cambridge Univ. Press.
  • [3] Billingsley, P. (1968)., Convergence of Probability Measures. New York: Wiley.
  • [4] Coles, S.G. (2001)., An Introduction to Statistical Modeling of Extreme Values. London: Springer.
  • [5] Coles, S.G. and Tawn, J.A. (1991). Modelling extreme multivariate events., J. Roy. Statist. Soc. Ser. B 53 377–392.
  • [6] Drees, H. and Huang, X. (1998). Best attainable rates of convergence for estimates of the stable tail dependence function., J. Multivariate Anal. 64 25–47.
  • [7] Einmahl, J.H.J., de Haan, L. and Li, D. (2006). Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition., Ann. Statist. 34 1987–2014.
  • [8] Falk, M., Hüsler, J. and Reiss, R.-D. (2004)., Laws of Small Numbers: Extremes and Rare Events. Basel: Birkhäuser.
  • [9] Fang, H.-B., Fang, K.-T. and Kotz, S. (2002). The meta-elliptical distributions with given marginals., J. Multivariate Anal. 82 1–16.
  • [10] Galambos, J. (1987)., The Asymptotic Theory of Extreme Order Statistics. Malaber: Krieger.
  • [11] Geluk, J.L., de Haan, L. and de Vries, C.G. (2007). Weak & strong financial fragility. Tinbergen Institute Discussion Paper, 2007-023/2.
  • [12] Genest, C., Favre, A.-C., Béliveau, J. and Jacques, C. (2007). Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data., Water Resources Research 43 W09401 DOI:10.1029/2006WR005275.
  • [13] de Haan, L. and Ferreira, A. (2006)., Extreme Value Theory: An Introduction. Berlin: Springer.
  • [14] de Haan, L., Neves, C. and Peng, L. (2008). Parametric tail copula estimation and model testing., J. Multivariate Anal. To appear.
  • [15] Huang, X. (1992). Statistics of bivariate extreme values. Ph.D. thesis, Erasmus Univ. Rotterdam, Tinbergen Institute Research Series, 22.
  • [16] Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions., Adv. in Appl. Probab. 34 587–608.
  • [17] Joe, H., Smith, R.L. and Weissman, I. (1992). Bivariate threshold methods for extremes., J. Roy. Statist. Soc. Ser. B 54 171–183.
  • [18] Klüppelberg, C., Kuhn, G. and Peng, L. (2007). Estimating tail dependence of elliptical distributions., Bernoulli 13 229–251.
  • [19] Klüppelberg, C., Kuhn, G. and Peng, L. (2008). Semi-parametric models for the multivariate tail dependence function – the asymptotically dependent case., Scand. J. Statist. To appear.
  • [20] Kotz, S. and Nadarajah, S. (2000)., Extreme Value Distributions. London: Imperial College Press.
  • [21] Ledford, A.W. and Tawn, J.A. (1996). Statistics for near independence in multivariate extreme values., Biometrika 83 169–187.
  • [22] Resnick, S. (1987)., Extreme Values, Regular Variation and Point Processes. New York: Springer.
  • [23] Smith, R.L. (1994). Multivariate threshold methods. In, Extreme Value Theory and Applications (J. Galambos, J. Lechner and E. Simiu, eds.) 225–248. Norwell: Kluwer.
  • [24] Tiago de Oliveira, J. (1980). Bivariate extremes: Foundations and statistics. In, Multivariate Analysis V (P.R. Krishnaiah, ed.) 349–366. Amsterdam: North-Holland.
  • [25] Tiago de Oliveira, J. (1989). Statistical decision for bivariate extremes. In, Extreme Value Theory: Proceedings, Oberwolfach 1987. Lecture Notes in Statist. (J. Hüsler and R.D. Reiss, eds.) 51 246–261. Berlin: Springer.
  • [26] van der Vaart, A.W. (1998)., Asymptotic Statistics. Cambridge: Cambridge Univ. Press.
  • [27] van der Vaart, A.W. and Wellner, J.A. (1996)., Weak Convergence and Empirical Processes. New York: Springer.
  • [28] Vervaat, W. (1972). Functional central limit theorems for processes with positive drift and their inverses., Z. Wahrsch. Verw. Gebiete 23 249–253.