Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Dependence Modeling

Ed. by Puccetti, Giovanni

1 Issue per year

Covered by:
WoS (ESCI)
SCOPUS
MathSciNet
zbMATH



Emerging Science

Open Access
Online
ISSN
2300-2298
See all formats and pricing
More options …

On an asymmetric extension of multivariate Archimedean copulas based on quadratic form

Elena Di Bernardino
  • Corresponding author
  • Elena Di Bernardino, CNAM, Paris, EA4629, Département IMATH, 292 rue Saint-Martin, Paris Cedex 03, France
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Didier Rullière
  • Corresponding author
  • Didier Rullière, Université de Lyon, Université Lyon 1, ISFA, Laboratoire SAF, 50 avenue Tony Garnier, 69366 Lyon, France
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2016-12-14 | DOI: https://doi.org/10.1515/demo-2016-0019

Abstract

An important topic in Quantitative Risk Management concerns the modeling of dependence among risk sources and in this regard Archimedean copulas appear to be very useful. However, they exhibit symmetry, which is not always consistent with patterns observed in real world data. We investigate extensions of the Archimedean copula family that make it possible to deal with asymmetry. Our extension is based on the observation that when applied to the copula the inverse function of the generator of an Archimedean copula can be expressed as a linear form of generator inverses. We propose to add a distortion term to this linear part, which leads to asymmetric copulas. Parameters of this new class of copulas are grouped within a matrix, thus facilitating some usual applications as level curve determination or estimation. Some choices such as sub-model stability help associating each parameter to one bivariate projection of the copula. We also give some admissibility conditions for the considered copulas. We propose different examples as some natural multivariate extensions of Farlie-Gumbel-Morgenstern or Gumbel-Barnett.

Keywords: Archimedean copulas; transformations of Archimedean copulas; quadratic form

References

  • [1] Alfonsi, A. and D. Brigo (2005). New families of copulas based on periodic functions. Comm. Statist. Theory Methods 34, 1437–1447. Google Scholar

  • [2] Billingsley, P. (1979). Probability and Measure. Wiley, New York. Google Scholar

  • [3] Brechmann, E. C. (2014). Hierarchical Kendall copulas: properties and inference. Canad. J. Statist. 42(1), 78–108. Google Scholar

  • [4] Capéraà, P., A.-L. Fougères, and C. Genest (2000). Bivariate distributions with given extreme value attractor. J. Multivariate Anal. 72(1), 30–49. Google Scholar

  • [5] Charpentier, A., A.-L. Fougères, C. Genest, and J. Nešlehová (2014). Multivariate Archimax copulas. J.Multivariate Anal. 126, 118–136. Web of ScienceGoogle Scholar

  • [6] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Acad. Roy. Belg. Bull. Cl. Sci. 65(5), 274–292. Google Scholar

  • [7] Di Bernardino, E. and D. Rullière (2013). Distortions of multivariate distribution functions and associated level curves: applications in multivariate risk theory. Insurance Math. Econom. 53(1), 190 – 205. Google Scholar

  • [8] Di Bernardino, E. and D. Rullière (2013). On certain transformations of Archimedean copulas: application to the nonparametric estimation of their generators. Depend. Model. 1, 1–36. Google Scholar

  • [9] Dolati, A. and M. Úbeda Flores (2006). Some new parametric families of multivariate copulas. Int.Math. Forum 1(1-4), 17–25. Google Scholar

  • [10] Durrleman, V., A. Nikeghbali, and T. Roncalli (2000). A simple transformation of copulas. Technical report, Groupe de Research Operationnelle Credit Lyonnais. Google Scholar

  • [11] Embrechts, P., F. Lindskog, and A. Mcneil (2003). Modelling dependence with copulas and applications to riskmanagement. In S. T. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance, pp. 329 – 384. North-Holland, Amsterdam. Google Scholar

  • [12] Hofert, M. (2010). Construction and sampling of nested archimedean copulas. In P. Jaworski, F. Durante, W. K. Härdle, and T. Rychlik (Eds.), Copula Theory and Its Applications, pp. 147–160. Springer, Berlin Heidelberg. Google Scholar

  • [13] Genest, C. and J. G. Nešlehová (2013). Assessing and modeling asymmetry in bivariate continuous data. In P. Jaworski, F. Durante, and K. W. Härdle (Eds.), Copulae in Mathematical and Quantitative Finance, pp. 91–114. Springer, Berlin Heidelberg. Google Scholar

  • [14] Hofert, M. (2008). Sampling Archimedean copulas. Comput. Statist. Data Anal. 52(12), 5163–5174. CrossrefGoogle Scholar

  • [15] Hofert, M. and M. Mächler (2011). Nested Archimedean copulas meet R: The nacopula package. J. Stat. Softw. 39(9), 1–20. Google Scholar

  • [16] Joe, H. (1990). Multivariate concordance. J. Multivariate Anal. 35(1), 12–30. Google Scholar

  • [17] Khoudraji, A. (1995). Contributions à l’étude des copules et à la modélisation de valeurs extrêmes bivariées. Ph.D. thesis, Université Laval, Québec, Canada. Google Scholar

  • [18] Kim, J.-M., S., E.A., T. Choi, and T. Heo (2011). Generalized bivariate copulas and their properties. Model. Assist. Stat. Appl. 6(2), 127–136. Google Scholar

  • [19] Klement, E. P., R. Mesiar, and E. Pap (2005). Transformations of copulas. Kybernetika 41(4), 425–434. Google Scholar

  • [20] Liebscher, E. (2008). Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99(10), 2234–2250. Web of ScienceGoogle Scholar

  • [21] McNeil, A. J. (2008). Sampling nested Archimedean copulas. J. Stat. Comput. Simul. 78(6), 567–581. CrossrefGoogle Scholar

  • [22] McNeil, A. and J. Nešlehová (2009). Multivariate Archimedean copulas, d-monotone functions and l1−norm symmetric distributions. Ann. Statist. 37(5B), 3059–3097. Google Scholar

  • [23] McNeil, A. J. and J. Nešlehová (2010). From Archimedean to Liouville copulas. J. Multivariate Anal. 101(8), 1772–1790. Google Scholar

  • [24] Mesiar, R. and V. Jágr (2013). d-Dimensional dependence functions and Archimax copulas. Fuzzy Sets Syst. 228, 78–87. Google Scholar

  • [25] Mesiar, R. and V. Najjari (2014). New families of symmetric/asymmetric copulas. Fuzzy Sets Syst. 252, 99–110. Google Scholar

  • [26] Morillas, P. M. (2005). A method to obtain new copulas from a given one. Metrika 61(2), 169–184. CrossrefGoogle Scholar

  • [27] Nelsen, R. B. (2006). An Introduction to Copulas. Second Edition. Springer, New York. Google Scholar

  • [28] Nelsen, R. B., J. J. Quesada-Molina, J. A. Rodríguez-Lallena, and M. Úbeda-Flores (2008). On the construction of copulas and quasi-copulas with given diagonal sections. Insurance Math. Econom. 42(2), 473–483. Google Scholar

  • [29] Rodríguez-Lallena, J. and M. Úbeda Flores (2004). A new class of bivariate copulas. Statist. Probab. Lett. 66(3), 315 – 325. CrossrefGoogle Scholar

  • [30] Sibuya, M. (1960). Bivariate extreme statistics. Ann. Inst. Statist. Math. 11, 195–210. CrossrefGoogle Scholar

  • [31] Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. de l’Institut de statistique de l’Université de Paris 8, 229–231. Google Scholar

  • [32] Valdez, E. and Y. Xiao (2011). On the distortion of a copula and its margins. Scand. Actuar. J. 4, 292–317. CrossrefWeb of ScienceGoogle Scholar

  • [33] Wu, S. (2014). Construction of asymmetric copulas and its application in two-dimensional reliability modelling. European J. Oper. Res. 238(2), 476 – 485. CrossrefGoogle Scholar

About the article

Received: 2016-09-14

Accepted: 2016-11-15

Published Online: 2016-12-14


Citation Information: Dependence Modeling, Volume 4, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2016-0019.

Export Citation

© 2016 Elena Di Bernardino and Didier Rullière. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in