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Using B-splines for nonparametric inference on bivariate extreme-value copulas

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Abstract

A visual tool is proposed for detecting the presence of extreme-value dependence or extremal tail behavior in bivariate data. The points appearing on the plot stem from rank-based transformations of the observations and can serve to estimate the unknown Pickands dependence function of the underlying extreme-value copula or its attractor. Quadratic constrained B-spline smoothing is used to derive an intrinsic estimator, which naturally leads to a test of extremeness. Both the estimator and the test are seen to perform well in simulations. The proposed methodology is illustrated with real data and the treatment of ties is briefly discussed.

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References

  • Ben Ghorbal, N., Genest, C., Nešlehová, J.: On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence. Canad. J. Statist. 37, 534–552 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Berghaus, B., Bücher, A., Dette, H.: Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence. J. Soc. Fr. Statist. 154, 116–137 (2013)

    Google Scholar 

  • Bücher, A., Dette, H., Volgushev, S.: New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39, 1963–2006 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Capéraà, P., Fougères, A.-L., Genest, C.: A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84, 567–577 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Coles, S., Heffernan, J., Tawn, J.: Dependence measures for extreme value analyses. Extremes 2, 339–365 (1999)

    Article  MATH  Google Scholar 

  • de Boor, C.: A comment on “Numerical comparisons of algorithms for polynomial and rational multivariate approximations”, by J.N. Henry, M.S. Henry and D. Schmidt. SIAM J. Numer. Anal. 15, 1208–1211 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  • Demarta, S., McNeil, A.J.: The t copula and related copulas. Internat. Statist. Rev. 73, 111–129 (2005)

    Article  MATH  Google Scholar 

  • Du, Y., Nešlehová, J.: A moment-based test for extreme-value dependence. Metrika 76, 673–695 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  • Einmahl, J.H.J., Segers, J.: Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution. Ann. Statist. 37, 2953–2989 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Feidt, A., Genest, C., Nešlehová, J.: Asymptotics of joint maxima for discontinuous random variables. Extremes 13, 35–53 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Fils-Villetard, A., Guillou, A., Segers, J.: Projection estimators of Pickands dependence functions. Canad. J. Statist. 36, 369–382 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  • Frees, E.W., Valdez, E.A.: Understanding relationships using copulas. N. Am. Actuar. J. 2, 1–25 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Garralda-Guillem, A.I.: Structure de dépendance des lois de valeurs extrêmes bivariées. C. R. Acad. Sci. Paris Sér. I Math. 330, 593–596 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  • Genest, C., Ghoudi, K., Rivest, L.-P.: Comment on the paper by E.W. Frees and E.A. Valdez entitled “Understanding relationships using copulas”. North Amer. Act. J. 2, 143–149 (1998)

    Article  MathSciNet  Google Scholar 

  • Genest, C., Nešlehová, J.: A primer on copulas for count data. Astin Bull. 37, 475–515 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Genest, C., Nešlehová, J.: Copula modeling for extremes, 2nd edn. In: El-Shaarawi, A.H., Piegorsch, W.W. (eds.) Encyclopedia of Environmetric, vol. 2, pp 530–541. Wiley, Chichester (2012)

    Google Scholar 

  • Genest, C., Nešlehová, J.: Copulas and copula models, 2nd edn. In: El-Shaarawi, A.H., Piegorsch, W.W. (eds.) Encyclopedia of Environmetrics, vol. 2, pp 541–553. Wiley, Chichester (2012)

    Google Scholar 

  • Genest, C., Nešlehová, J.G.: Modeling dependence beyond correlation. In: Lawless, J.F. (ed.) Statistics in Action: A Canadian Outlook, pp 59–78. Chapman & Hall, London (2014)

    Google Scholar 

  • Genest, C., Nešlehová, J., Ruppert, M.: Comment on the paper by S. Haug, C. Klüppelberg and L. Peng entitled “Statistical models and methods for dependence in insurance data”. J. Korean Statist. Soc. 40, 141–148 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Genest, C., Segers, J.: Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37, 2990–3022 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Ghoudi, K., Khoudraji, A., Rivest, L.-P.: Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Canad. J. Statist. 26, 187–197 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Haug, S., Klüppelberg, C., Peng, L.: Statistical models and methods for dependence in insurance data. J. Korean Statist. Soc. 40, 125–139 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • He, X., Ng, P.: COBS: Qualitatively constrained smoothing via linear programming. Comput. Statist. 14, 315–337 (1999)

    Article  MATH  Google Scholar 

  • Kammerer, W., Reddien, G., Varga, R.: Quadratic interpolatory splines. Numer. Math. 22, 241–259 (2007)

    Article  MathSciNet  Google Scholar 

  • Koenker, R., Ng, P., Portnoy, S.: Quantile smoothing splines. Biometrika 81, 673–680 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  • Kojadinovic, I., Segers, J., Yan, J.: Large-sample tests of extreme-value dependence for multivariate copulas. Canad. J. Statist. 39, 703–720 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Kojadinovic, I., Yan, J.: Nonparametric rank-based tests of bivariate extreme-value dependence. J. Multivariate Anal. 101, 2234–2249 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Ledford, A.W., Tawn, J.A.: Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  • Lee, J.S., Cox, D.D.: Robust smoothing: smoothing parameter selection and applications to fluorescence spectroscopy. Comput. Statist. Data Anal. 54, 3131–3143 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • McNeil, A.J.: Estimating the tails of loss severity distributions using extreme value theory. Astin Bull. 27, 117–137 (1997)

    Article  Google Scholar 

  • McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton (2005)

    Google Scholar 

  • Ng, P., Mächler, M.: A fast and efficient implementation of qualitatively constrained quantile smoothing splines. Stat. Model. 7, 315–328 (2007)

    Article  MathSciNet  Google Scholar 

  • Pickands, J.: Multivariate extreme value distributions (with discussion). In: Proceedings of the 43rd Session of the International Statistical Institute, vol. 2, pp. 859–878, 894–902. Buenos Aires (1981)

  • Quessy, J.-F.: Testing for bivariate extreme dependence using Kendall’s process. Scand. J. Statist. 39, 497–514 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Ramsay, J.O.: Monotone regression splines in action (with discussion). Statist. Sci. 3, 425–441 (1988)

    Article  Google Scholar 

  • Ruppert, D.: Selecting the number of knots for penalized splines. J. Comput. Graph. Stat. 11, 735–757 (2002)

    Article  MathSciNet  Google Scholar 

  • Segers, J.: Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18, 764–782 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Tawn, J.A.: Bivariate extreme value theory: Models and estimation. Biometrika 75, 397–415 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Cormier, E., Genest, C. & Nešlehová, J.G. Using B-splines for nonparametric inference on bivariate extreme-value copulas. Extremes 17, 633–659 (2014). https://doi.org/10.1007/s10687-014-0199-4

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  • DOI: https://doi.org/10.1007/s10687-014-0199-4

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