Rank-based inference for bivariate extreme-value copulas. (English) Zbl 1173.62013

Summary: Consider a continuous random pair \((X,Y)\) whose dependence is characterized by an extreme-value copula with Pickands dependence function \(A\). When the marginal distributions of \(X\) and \(Y\) are known, several consistent estimators of \(A\) are available. Most of them are variants of the estimators due to J. Pickands [Bull. Int. Stat. Inst. 49, 859–878 (1981; Zbl 0518.62045)] and P. Capéraà, A.-L. Fougères and C. Genest [Biometrika 84, No. 3, 567–577 (1997; Zbl 1058.62516)]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of \(X\) and \(Y\) are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.


62G05 Nonparametric estimation
62G32 Statistics of extreme values; tail inference
62G20 Asymptotic properties of nonparametric inference
65C05 Monte Carlo methods
Full Text: DOI arXiv


[1] Abdous, B. and Ghoudi, K. (2005). Non-parametric estimators of multivariate extreme dependence functions. J. Nonparametr. Statist. 17 915-935. · Zbl 1080.62027
[2] Beirlant, J., Goegebeur, Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes : Theory and Applications . Wiley, Chichester. · Zbl 1070.62036
[3] Capéraà, P., Fougères, A.-L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567-577. · Zbl 1058.62516
[4] Cebrián, A. C., Denuit, M. and Lambert, P. (2003). Analysis of bivariate tail dependence using extreme value copulas: An application to the SOA medical large claims database. Belgian Actuarial Bulletin 3 33-41.
[5] Coles, S. G. and Tawn, J. A. (1994). Statistical methods for multivariate extremes: An application to structural design. J. Appl. Statist. 43 1-48. · Zbl 0825.62717
[6] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. ( 5 ) 65 274-292. · Zbl 0422.62037
[7] Deheuvels, P. (1984). Probabilistic aspects of multivariate extremes. In Statistical Extremes and Applications (J. Tiago de Oliveira, ed.) 117-130. Reidel, Dordrecht. · Zbl 0562.62015
[8] Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Lett. 12 429-439. · Zbl 0749.62033
[9] Demarta, S. and McNeil, A. J. (2005). The t copula and related copulas. Internat. Statist. Rev. 73 111-129. · Zbl 1104.62060
[10] Dudley, R. M. and Koltchinskii, V. I. (1994). Envelope moment conditions and Donsker classes. Teor. Ĭmovīr. Mat. Stat. 51 39-49. · Zbl 0941.60059
[11] Fermanian, J.-D., Radulović, D. and Wegkamp, M. H. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847-860. · Zbl 1068.62059
[12] Fils-Villetard, A., Guillou, A. and Segers, J. (2008). Projection estimators of Pickands dependence functions. Canad. J. Statist. 36 369-382. · Zbl 1153.62044
[13] Galambos, J. (1978). The Asymptotic Theory of Extreme Order Statistics . Wiley, New York. · Zbl 0381.62039
[14] Ghoudi, K., Khoudraji, A. and Rivest, L.-P. (1998). Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canad. J. Statist. 26 187-197. · Zbl 0899.62071
[15] Gumbel, É. J. (1960). Distributions des valeurs extrêmes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris 9 171-173. · Zbl 0093.15303
[16] Hall, P. and Tajvidi, N. (2000). Distribution and dependence-function estimation for bivariate extreme-value distributions. Bernoulli 6 835-844. · Zbl 1067.62540
[17] Henmi, M. (2004). A paradoxical effect of nuisance parameters on efficiency of estimators. J. Japan Statist. Soc. 34 75-86. · Zbl 1061.62053
[18] Hsing, T. (1989). Extreme value theory for multivariate stationary sequences. J. Multivariate Anal. 29 274-291. · Zbl 0679.62039
[19] Hüsler, J. and Reiss, R.-D. (1989). Maxima of normal random vectors: Between independence and complete dependence. Statist. Probab. Lett. 7 283-286. · Zbl 0679.62038
[20] Jiménez, J. R., Villa-Diharce, E. and Flores, M. (2001). Nonparametric estimation of the dependence function in bivariate extreme value distributions. J. Multivariate Anal. 76 159-191. · Zbl 0998.62050
[21] Joe, H. (1990). Families of min-stable multivariate exponential and multivariate extreme value distributions. Statist. Probab. Lett. 9 75-81. · Zbl 0686.62035
[22] Joe, H., Smith, R. L. and Weissman, I. (1992). Bivariate threshold methods for extremes. J. Roy. Statist. Soc. Ser. B 54 171-183. JSTOR: · Zbl 0775.62083
[23] Marshall, A. W. (1970). Discussion of Barlow and van Zwet’s papers. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 175-176. Cambridge Univ. Press, London.
[24] Pickands, J. (1981). Multivariate extreme value distributions (with a discussion). In Proceedings of the 43rd Session of the International Statistical Institute. Bull. Inst. Internat. Statist. 49 859-878, 894-902. · Zbl 0518.62045
[25] Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Statist. 4 912-923. · Zbl 0359.62040
[26] 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. Nova Science Publishers, New York.
[27] Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229-231. · Zbl 0100.14202
[28] Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. Ann. Probab. 12 361-379. · Zbl 0533.62037
[29] Tawn, J. A. (1988). Bivariate extreme value theory: Models and estimation. Biometrika 75 397-415. JSTOR: · Zbl 0653.62045
[30] Tiago de Oliveira, J. (1980). Bivariate extremes: Foundations and statistics. In Multivariate Analysis V (P. R. Krishnaiah, ed.) 349-366. North-Holland, Amsterdam. · Zbl 0431.62029
[31] Tsukahara, H. (2005). Semiparametric estimation in copula models. Canad. J. Statist. 33 357-375. · Zbl 1077.62022
[32] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002
[33] Zhang, D., Wells, M. T. and Peng, L. (2008). Nonparametric estimation of the dependence function for a multivariate extreme value distribution. J. Multivariate Anal. 99 577-588. · Zbl 1333.62140
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.