Estimating multivariate extremal dependence: a new proposal
Author:
M. Ferreira
Journal:
Theor. Probability and Math. Statist. 93 (2016), 169-175
MSC (2010):
Primary 62G05; Secondary 62G32
DOI:
https://doi.org/10.1090/tpms/1001
Published electronically:
February 7, 2017
MathSciNet review:
3553448
Full-text PDF Free Access
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Additional Information
Abstract: Multivariate extreme values require the use of extreme-value copulas, as they appear in the limit of componentwise maxima. These can be characterized by the so-called Pickands dependence function. A new multivariate nonparametric estimator will be presented, along with convergence properties. Based on simulations, we will analyze its performance and compare with well-known estimators from the literature.
References
- Belkacem Abdous and Kilani Ghoudi, Non-parametric estimators of multivariate extreme dependence functions, J. Nonparametr. Stat. 17 (2005), no. 8, 915–935. MR 2192166, DOI 10.1080/10485250500336379
- Jan Beirlant, Yuri Goegebeur, Jozef Teugels, and Johan Segers, Statistics of extremes, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2004. Theory and applications; With contributions from Daniel De Waal and Chris Ferro. MR 2108013, DOI 10.1002/0470012382
- Axel Bücher, Holger Dette, and Stanislav Volgushev, New estimators of the Pickands dependence function and a test for extreme-value dependence, Ann. Statist. 39 (2011), no. 4, 1963–2006. MR 2893858, DOI 10.1214/11-AOS890
- P. Capéraà, A.-L. Fougères, and C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997), 567–577.
- Paul Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991), no. 5, 429–439. MR 1142097, DOI 10.1016/0167-7152(91)90032-M
- Helena Ferreira and Marta Ferreira, On extremal dependence of block vectors, Kybernetika (Prague) 48 (2012), no. 5, 988–1006. MR 3086865
- M. Ferreira, A new estimator for the Pickands dependence function (2015). (submitted)
- Amélie Fils-Villetard, Armelle Guillou, and Johan Segers, Projection estimators of Pickands dependence functions, Canad. J. Statist. 36 (2008), no. 3, 369–382 (English, with English and French summaries). MR 2456011, DOI 10.1002/cjs.5550360303
- Christian Genest and Johan Segers, Rank-based inference for bivariate extreme-value copulas, Ann. Statist. 37 (2009), no. 5B, 2990–3022. MR 2541453, DOI 10.1214/08-AOS672
- Gordon Gudendorf and Johan Segers, Nonparametric estimation of an extreme-value copula in arbitrary dimensions, J. Multivariate Anal. 102 (2011), no. 1, 37–47. MR 2729418, DOI 10.1016/j.jmva.2010.07.011
- Gordon Gudendorf and Johan Segers, Nonparametric estimation of multivariate extreme-value copulas, J. Statist. Plann. Inference 142 (2012), no. 12, 3073–3085. MR 2956794, DOI 10.1016/j.jspi.2012.05.007
- Peter Hall and Nader Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000), no. 5, 835–844. MR 1791904, DOI 10.2307/3318758
- James Pickands III, Multivariate extreme value distributions, Proceedings of the 43rd session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981), 1981, pp. 859–878, 894–902 (English, with French summary). With a discussion. MR 820979
- J. Segers, Nonparametric inference for bivariate extreme-value copulas, Topics in Extreme Values (M. Ahsanullah and S. N. U. A. Kirmani, eds.), Nova Science Publishers, New York, 2007, 181–203.
- Johan Segers, Asymptotics of empirical copula processes under non-restrictive smoothness assumptions, Bernoulli 18 (2012), no. 3, 764–782. MR 2948900, DOI 10.3150/11-BEJ387
- J. A. Tawn, Modelling multivariate extreme value distributions, Biometrika 77 (1990), no. 2, 245–253.
- Aad W. van der Vaart and Jon A. Wellner, Weak convergence and empirical processes, Springer Series in Statistics, Springer-Verlag, New York, 1996. With applications to statistics. MR 1385671, DOI 10.1007/978-1-4757-2545-2
- Dabao Zhang, Martin T. Wells, and Liang Peng, Nonparametric estimation of the dependence function for a multivariate extreme value distribution, J. Multivariate Anal. 99 (2008), no. 4, 577–588. MR 2406072, DOI 10.1016/j.jmva.2006.09.011
References
- B. Abdous and K. Ghoudi, Non-parametric estimators of multivariate extreme dependence functions, J. Nonparametr. Statist. 17 (2005), 915–935. MR 2192166
- J. Beirlant, Y. Goegebeur, J. Segers, and J. Teugels, Statistics of Extremes: Theory and Applications, Wiley, Chichester, 2004. MR 2108013
- A. Bücher, H. Dette, and S. Volgushev, New estimators of the Pickands dependence function and a test for extreme-value dependence, Ann. Statist. 39 (2011), no. 4, 1963–2006. MR 2893858
- P. Capéraà, A.-L. Fougères, and C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997), 567–577.
- P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme value distributions, Statist. Probab. Lett. 12 (1991), 429–439. MR 1142097
- H. Ferreira and M. Ferreira, On extremal dependence of block vectors, Kybernetika 48 (2012), no. 5, 988–1006. MR 3086865
- M. Ferreira, A new estimator for the Pickands dependence function (2015). (submitted)
- A. Fils-Villetard, A. Guillou, and J. Segers, Projection estimators of Pickands dependence functions, Canad. J. Statist. 36 (2008), 369–382. MR 2456011
- C. Genest and J. Segers, Rank-based inference for bivariate extreme-value copulas, Ann. Statist. 37 (2009), no. 5B, 2990–3022. MR 2541453
- G. Gudendorf and J. Segers, Nonparametric estimation of an extreme-value copula in arbitrary dimensions, J. Multivariate Anal. 102 (2011), no. 1, 37–47. MR 2729418
- G. Gudendorf and J. Segers, Nonparametric estimation of multivariate extreme-value copulas, J. Statist. Plann. Inference 142 (2012), 3073–3085. MR 2956794
- P. Hall and N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000), 835–844. MR 1791904
- J. Pickands, Multivariate extreme value distributions (with a discussion), Proceedings of the 43rd Session of the International Statistical Institute, Bull. Inst. Internat. Statist., vol. 49, 1981, pp. 859–878, 894–902. MR 820979
- J. Segers, Nonparametric inference for bivariate extreme-value copulas, Topics in Extreme Values (M. Ahsanullah and S. N. U. A. Kirmani, eds.), Nova Science Publishers, New York, 2007, 181–203.
- J. Segers, Asymptotics of empirical copula processes under non-restrictive smoothness assumptions, Bernoulli 18 (2012), no. 3, 764–782. MR 2948900
- J. A. Tawn, Modelling multivariate extreme value distributions, Biometrika 77 (1990), no. 2, 245–253.
- A. W. van der Vaart and J. A. Wellner, Weak Convergence and Empirical Processes, Springer, New York, 1996. MR 1385671
- D. Zhang, M. T. Wells, and L. Peng, Nonparametric estimation of the dependence function for a multivariate extreme value distribution, J. Multivariate Anal. 99 (2008), 577–588. MR 2406072
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Additional Information
M. Ferreira
Affiliation:
Center of Mathematics of University of Minho, Campus de Gualtar, Braga, Portugal
Address at time of publication:
CEMAT (Center for Computational and Stochastic Mathematics) of Instituto Superior Técnico, University of Lisbon, Portugal
Email:
msferreira@math.uminho.pt
Keywords:
Extreme value copula,
multivariate Pickands dependence function,
nonparametric estimation
Received by editor(s):
July 31, 2015
Published electronically:
February 7, 2017
Article copyright:
© Copyright 2017
American Mathematical Society