A Graphical Tool for Extreme Value Copula Selection Based on the Pickands Dependence Function

Year 2022, Volume: 14 Issue: 2, 38 - 50, 31.12.2022

Selim Orhun Susam

Abstract

We present a graphical tool that was primarily proposed by Michiels et al. [18] and later modified by Durante et al. [4]. We also improve this method to select the better fit of the given data among some extreme value copulas based on the Pickands dependence function. We conduct a Monte Carlo simulation study to investigate its performance. Also, the graphical method is illustrated by a real data example.

Keywords

Copula, Pickands function, Extreme value copula

References

• Ahmadabadi, A. and Ucer, B.H. (2017). Bivariate nonparametric estimation of the Pickands dependence function using Bernstein copula with kernel regression approach. Computitional Statistics, 32, 1515-1532.
• Brechmann, E.C. (2013). Properties of extreme-value copulas. [Thesis, Technical University of Munich].
• Capéraà, P., Fougres, A.L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567-577.
• Durante, F., Fernández-Sánchez, J. and Pappad´a, R. (2015). Copulas, diagonals and tail dependence. Fuzzy Sets and Systems, 264, 22-41.
• Dutfoy, A., Parey, S. and Roche, N. (2017). Multivariate extreme value theory-A tutorial with applications to hydrology and meteorology. Dependence Modeling, 2, 30-48.
• Frees, E. and Valdez, E. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2, 1-25.
• Galambos, J. (1975). Order statistics of samples from multivariate distributions. Journal of the American Statistical Association, 70, 674-680.
• Genest, C., Kojadinovic, I., Néslehová, J. and Yan, J. (2011). A goodness-of-fit test for bivariate extremevalue copulas. Bernoulli, 17(1), 253-275.
• Genest, C. and Segers, J. (2009). Rank-Based inference for bivariate extreme value copulas. The Annals of Statistics, 37, 2990-3022.
• Gudendorf, G. and Segers, J. (2011). Nonparametric estimation of an extreme-value copula in arbitrary dimensions. Journal of Multivariate Analysis, 102(1), 37-47.
• Guillotte, S. and Perron, F. (2016). Polynomial pickands functions. Bernoulli, 22(1), 213-241.
• Gumbel, E.J. (1960). Distributios des valeurs extr´emes en plusiers dimensions. Publications de l’Institut de Statistique de l’Université de Paris, 9, 171-173.
• Hougaard, P. (1986). A class of multivariate failure time distributions. Biometrika, 73, 671-678.
• Husler, J. (1986). Extreme values of non-stationary random sequences. Journal of Applied Probability, 23, 937-950.
• Joe, H. (1990). Multivariate concordance. Journal of Multivariate Analysis, 35, 12-30.
• Marcon, G., Padoan, S.A. and Antoniano-Villalobos, I. (2016). Bayesian inference for the extremal dependence. Electronic Journal of Statistics, 10(2), 3310-3337.
• Marcon, G., Padoan, S.A., Naveau, P., Muliere, P. and Segers, J. (2017). Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials. Journal of Statistical Planning and Inference, 183, 1-17.
• Michiels, F. and De Schepper, A. (2013). A new graphical tool for copula selection. Journal of Computational and Graphical Statistics, 22(2), 471-493.
• Pickands, J. (1981). Multivariate extreme value distribution. In Proceedings of the International Statistical Institute, 859-878.
• Sklar, A. (1959). Fonctions de repartition a n dimensions et leurs marges. Publications de lInstitut de Statistique de lUniversite de Paris, 8, 229-231.
• Vettori, S., Huser, R. and Genton, M.G. (2018). A comparison of dependence function estimators in multivariate extremes. Statistics and Computing, 28(3), 525-538.