Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence

Berghaus, Betina; Bücher, Axel; Dette, Holger

Journal de la société française de statistique, Tome 154 (2013) no. 1, pp. 116-137.

Nous nous intéressons à l’estimation de la fonction de dépendance de Pickands correspondant à une distribution de valeurs extrêmes multivariée. Un estimateur du minimum de distance fondé sur la distance $L^2$ entre les logarithmes de la copule empirique et de la copule inconnue est proposé et sa convergence faible est démontrée. Contrairement à d’autres procédures récemment proposées dans la littérature pour l’estimation de la fonction de dépendance de Pickands multivariée (voir [ Zhang et al., 2008 ] et [ Gudendorf and Segers, 2011 ]), les estimateurs étudiés dans ce travail ne requièrent pas la donnée des distributions marginales et sont ainsi une alternative à la méthode de [ Gudendorf and Segers, 2012 ]. De plus, l’approche du minimum de distance considérée permet naturellement la construction d’un test d’appartenance à la classe des copules de valeurs extrêmes dont la consistance est démontrée pour les copules modélisant une association positive. Des simulations sont enfin utilisées pour étudier empiriquement, sur des échantillons de taille finie, les propriétés de l’estimateur du minimum de distance ainsi que du test associé mis en oeuvre à l’aide d’un rééchantillonnage fondé sur des multiplicateurs.

We consider the problem of estimating the Pickands dependence function corresponding to a multivariate extreme-value distribution. A minimum distance estimator is proposed which is based on an $L^2$-distance between the logarithms of the empirical and the unknown extreme-value copula. The minimizer can be expressed explicitly as a linear functional of the logarithm of the empirical copula and weak convergence of the corresponding process on the simplex is proved. In contrast to other procedures which have recently been proposed in the literature for the nonparametric estimation of a multivariate Pickands dependence function (see [ Zhang et al., 2008 ] and [ Gudendorf and Segers, 2011 ]), the estimators constructed in this paper do not require knowledge of the marginal distributions and are an alternative to the method which has recently been suggested in [ Gudendorf and Segers, 2012 ]. Moreover, the minimum distance approach allows the construction of a simple test for the hypothesis of a multivariate extreme-value copula, which is consistent against a broad class of alternatives. The finite-sample properties of the estimator and a multiplier bootstrap version of the test are investigated by means of a simulation study.

Keywords: Extreme-value copula, minimum distance estimation, Pickands dependence function, weak convergence, empirical copula process
Mot clés : copules de valeurs extrêmes, estimateurs du minimum de distance, fonction de dépendance de Pickands, convergence faible, processus de copule empirique

[1] Bücher, Axel; Dette, Holger A note on bootstrap approximations for the empirical copula process, Statist. Probab. Lett., Volume 80 (2010), pp. 1925-1932 | Zbl

[2] Bücher, Axel; Dette, Holger; Volgushev, Stanislav New estimators of the Pickands dependence function and a test for extreme-value dependence, Ann. Statist., Volume 39 (2011), pp. 1963-2006 | DOI | MR | Zbl

[3] Ben Ghorbal, Noomen; Genest, Christian; Nešlehová, Johanna On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence, Canad. J. Statist., Volume 37 (2009) no. 4, pp. 534-552 | DOI | MR | Zbl

[4] Beirlant, Jan; Goegebeur, Yuri; Teugels, Jozef; Segers, Johan Statistics of extremes, Wiley Series in Probability and Statistics, John Wiley & Sons Ltd., Chichester, 2004, xiv+490 pages (Theory and applications, With contributions from Daniel De Waal and Chris Ferro) | DOI | MR | Zbl

[5] Bücher, Axel; Ruppert, Martin Consistent testing for a constant copula under strong mixing based on the tapered block multiplier technique, J. Multivariate Anal., Volume 116 (2013), pp. 208-229 | DOI | Zbl

[6] Bücher, Axel; Volgushev, Stanislav Empirical and sequential empirical copula processes under serial dependence, J. Multivariate Anal., Volume 119 (2013), pp. 61-70 | Zbl

[7] Cebrian, Ana C.; Denuit, Michel; Lambert, Philippe Analysis of bivariate tail dependence using extreme values copulas: An application to the SOA medical large claims database., Belgian Actuarial Journal, Volume 3 (2003), pp. 33-41

[8] Capéraà, Philippe; Fougères, Anne-Laure; Genest, Christian A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika, Volume 84 (1997) no. 3, pp. 567-577 | DOI | MR | Zbl

[9] Coles, Stuart; Heffernan, Janet; Tawn, Jonathan Dependence measures for extreme value analyses, Extremes, Volume 2 (1999), pp. 339-365 | Zbl

[10] Cook, R. Dennis; Johnson, Mark E. Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set, Technometrics, Volume 28 (1986) no. 2, pp. 123-131 | DOI | MR

[11] Deheuvels, Paul Probabilistic aspects of multivariate extremes, Statistical Extremes and Applications (de Oliveira, J. Tiago, ed.), Reidel, Dordrecht, 1984 | Zbl

[12] Deheuvels, Paul On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett., Volume 12 (1991), pp. 429-439 | Zbl

[13] Doukhan, Paul; Fermanian, Jean-David; Lang, Gabriel An empirical central limit theorem with applications to copulas under weak dependence, Stat. Inference Stoch. Process., Volume 12 (2009), pp. 65-87 | DOI | MR | Zbl

[14] Du, Yeting; Nešlehová, Johanna A moment-based test for extreme-value dependence, Metrika, to appear (2012) | DOI | Zbl

[15] Fermanian, Jean-David; Radulović, D.; Wegkamp, M. J. Weak convergence of empirical copula processes., Bernoulli, Volume 10 (2004), pp. 847-860 | Zbl

[16] Fils-Villetard, Amélie; Guillou, Armelle; Segers, Johan Projection estimators of Pickands dependence functions, Canad. J. Statist., Volume 36 (2008) no. 3, pp. 369-382 | DOI | MR | Zbl

[17] Genest, Christian; Kojadinovic, Ivan; Nešlehová, Johanna; Yan, Jun A goodness-of-fit test for bivariate extreme-value copulas, Bernoulli, Volume 17 (2011) no. 1, pp. 253-275 | DOI | MR | Zbl

[18] Ghoudi, Kilani; Khoudraji, Abdelhaq; Rivest, Louis-Paul Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles, Canad. J. Statist., Volume 26 (1998) no. 1, pp. 187-197 | DOI | MR | Zbl

[19] Genest, Christian; Segers, Johan Rank-based inference for bivariate extreme-value copulas, Ann. Statist., Volume 37 (2009), pp. 2990-3022 | Zbl

[20] Gudendorf, Gordon; Segers, Johan Nonparametric estimation of an extreme-value copula in arbitrary dimensions, J. Multivariate Anal., Volume 102 (2011), pp. 37-47 | Zbl

[21] Gudendorf, Gordon; Segers, Johan Nonparametric estimation of multivariate extreme-value copulas, J. Statist. Plann. Inference, Volume 142 (2012) no. 12, pp. 3073-3085 | DOI | MR | Zbl

[22] Hsing, Tailen Extreme value theory for multivariate stationary sequences, J. Multivariate Anal., Volume 29 (1989) no. 2, pp. 274-291 | DOI | MR | Zbl

[23] Hall, Peter; Tajvidi, Nader Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli, Volume 6 (2000) no. 5, pp. 835-844 | DOI | MR | Zbl

[24] Kosorok, Michael R. Introduction to Empirical Processes and Semiparametric Inference, Springer Series in Statistics, New York, 2008 | Zbl

[25] Kojadinovic, Ivan; Segers, Johan; Yan, Jun Large-sample tests of extreme-value dependence for multivariate copulas, Canad. J. Statist., Volume 39 (2011), pp. 703-720 | DOI | MR | Zbl

[26] Kojadinovic, Ivan; Yan, Jun Modeling multivariate distributions with continuous margins using the copula R package, Journal of Statistical Software, Volume 34 (2010) no. 9, pp. 1-20

[27] Kojadinovic, Ivan; Yan, Jun Nonparametric rank-based tests of bivariate extreme-value dependence, J. Multivariate Anal., Volume 101 (2010), pp. 2234-2249 | DOI | MR | Zbl

[28] Marshall, A. W. Discussion of Barlow and van Zwetís papers, Nonparametric Techniques in Statistical Inference (Pudi, M. L., ed.), Cambridge University Press, London, 1970, pp. 175-176

[29] McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul Quantitative risk management, Princeton Series in Finance, Princeton University Press, Princeton, NJ, 2005, xvi+538 pages (Concepts, techniques and tools) | MR | Zbl

[30] Pickands, James III Multivariate extreme value distributions, Proceedings of the 43rd session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981), Volume 49 (1981), p. 859-878, 894–902 (With a discussion) | MR | Zbl

[31] Quessy, Jean-François Testing for bivariate extreme dependence using Kendall’s process, Scand. J. Stat., Volume 39 (2012) no. 3, pp. 497-514 | DOI | Zbl

[32] Rojo Jiménez, Javier; Villa-Diharce, Enrique; Flores, Miguel Nonparametric estimation of the dependence function in bivariate extreme value distributions, J. Multivariate Anal., Volume 76 (2001) no. 2, pp. 159-191 | DOI | MR | Zbl

[33] Rémillard, Bruno; Scaillet, Olivier Testing for equality between two copulas, J. Multivariate Anal., Volume 100 (2009), pp. 377-386 | Zbl

[34] Robertson, Tim; Wright, Farroll T.; Dykstra, Richard L. Ordered Restricted Statistical Inference, Wiley, New York, 1996 | Zbl

[35] Rüschendorf, Ludger Asymptotic distributions of multivariate rank order statistics, Ann. Statist., Volume 4 (1976) no. 5, pp. 912-923 | MR | Zbl

[36] Segers, Johan Nonparametric inference for bivariate extreme-value copulas, Topics in Extreme Values (Ahsanullah, M.; Kirmani, S. N. U. A., eds.), Nova Science Publishers, New York, 2007

[37] Segers, Johan Asymptotics of empirical copula processes under non-restrictive smoothness assumptions, Bernoulli, Volume 18 (2012), pp. 764-782 | Zbl

[38] Sklar, Abe Fonctions de répartition à $n$ dimensions et leurs marges., Publ. Inst. Statist. Univ. Paris, Volume 8 (1959), pp. 229-231 | Zbl

[39] Stephenson, Alec evd: Extreme Value Distributions, R News, Volume 2 (2002) http://CRAN.R-project.org/doc/Rnews/ | Zbl

[40] Stephenson, Alec Simulating multivariate extreme value distributions of logistic type, Extremes, Volume 6 (2003), pp. 49-59 | DOI | MR | Zbl

[41] Tawn, Jonathan Bivariate extreme value theory: Models and estimation, Biometrika, Volume 75 (1988), pp. 397-415 | Zbl

[42] Tawn, Jonathan Modelling multivariate extreme value distributions, Biometrika, Volume 77 (1990), pp. 245-253 | Zbl

[43] van der Vaart, Aad W.; Wellner, Jon A. Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996, xvi+508 pages (With applications to statistics) | MR | Zbl

[44] Wang, Jane-Ling Asymptotically minimax estimators for distributions with increasing failure rate, Ann. Statist., Volume 44 (1986), pp. 1113-1131 | Zbl

[45] Yue, Sheng The Gumbel logistic model for representing a multivariate storm event, Advances in Water Resources, Volume 24 (2000) no. 2, pp. 179 -185 | DOI

[46] Zhang, Dabao; Wells, Martin T.; Peng, Liang Nonparametric estimation of the dependence function for a multivariate extreme value distribution, J. Multivariate Anal., Volume 99 (2008), pp. 577-588 | DOI | MR | Zbl