Advertisement

Classes and Families

  • Marius Hofert
  • Ivan Kojadinovic
  • Martin Mächler
  • Jun Yan
Chapter
Part of the Use R! book series (USE R)

Abstract

This chapter introduces the main copula classes and the corresponding sampling procedures, along with some copula transformations that are important for practical purposes.

References

  1. Beirlant, J., Goegebeur, Y., Segers, J., & Teugels, J. (2004). Statistics of extremes: Theory and applications. Wiley series in probability and statistics. Chichester: Wiley.CrossRefGoogle Scholar
  2. Bolker, B. (2017). bbmle: Tools for general maximum likelihood estimation. R package version 1.0.20. https://CRAN.R-project.org/package=bbmle.
  3. Capéraà, P., Fougères, A.-L., & Genest, C. (2000). Bivariate distributions with given extreme value attractor. Journal of Multivariate Analysis, 72, 30–49.MathSciNetCrossRefGoogle Scholar
  4. Charpentier, A., Fougères, A.-L., Genest, C., & Nešlehová, J. G. (2014). Multivariate Archimax copulas. Journal of Multivariate Analysis, 126, 118–136.MathSciNetCrossRefGoogle Scholar
  5. De Haan, L., & Ferreira, A. (2006). Extreme value theory: An introduction. New York: Springer.CrossRefGoogle Scholar
  6. Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review, 73(1), 111–129.CrossRefGoogle Scholar
  7. Devroye, L. (1993). A triptych of discrete distributions related to the stable law. Statistics & Probability Letters, 18(5), 349–351.MathSciNetCrossRefGoogle Scholar
  8. Fang, K.-T., Kotz, S., & Ng, K.-W. (1990). Symmetric multivariate and related distributions. London: Chapman & Hall.CrossRefGoogle Scholar
  9. Feller, W. (1971). An introduction to probability theory and its applications (2nd ed.), vol. 2. New York: Wiley.Google Scholar
  10. Galambos, J. (1978). The asymptotic theory of extreme order statistics. Wiley series in probability and mathematical statistics. New York: Wiley.Google Scholar
  11. Genest, C., Ghoudi, K., & Rivest, L.-P. (1998). Discussion of “Understanding relationships using copulas”, by E. Frees and E. Valdez. North American Actuarial Journal, 3, 143–149.CrossRefGoogle Scholar
  12. Genest, C., Kojadinovic, I., Nešlehová, J. G., & Yan, J. (2011a). A goodness-of-fit test for bivariate extreme-value copulas. Bernoulli, 17(1), 253–275.MathSciNetCrossRefGoogle Scholar
  13. Genest, C., & Nešlehová, J. G. (2013). Assessing and modeling asymmetry in bivariate continuous data. In P. Jaworski, F. Durante & W. K. Härdle (Eds.), Copulae in mathematical and quantitative finance (pp. 91–114). Lecture notes in statistics. Berlin: Springer.CrossRefGoogle Scholar
  14. Genz, A., Bretz, F., Miwa, T., Mi, X., Leisch, F., & Scheipl, F. (2017). mvtnorm: Multivariate normal and t distribution. R package version 1.0-6. https://CRAN.R-project.org/package=mvtnorm.
  15. Ghoudi, K., Khoudraji, A., & Rivest, L.-P. (1998). Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. The Canadian Journal of Statistics, 26(1), 187–197.MathSciNetCrossRefGoogle Scholar
  16. Gudendorf, G., & Segers, J. (2010), Extreme-value copulas. In P. Jaworski, F. Durante, W. K. Härdle & W. Rychlik (Eds.), Copula theory and its applications (Warsaw, 2009) (pp. 127–146). Lecture notes in statistics. Berlin: Springer.CrossRefGoogle Scholar
  17. Gudendorf, G., & Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. Journal of Statistical Planning and Inference, 143, 3073–3085.MathSciNetCrossRefGoogle Scholar
  18. Gumbel, E. J. (1958). Statistics of extremes. New York: Columbia University Press.zbMATHGoogle Scholar
  19. Hofert, M. (2008). Sampling Archimedean copulas. Computational Statistics & Data Analysis, 52, 5163–5174.MathSciNetCrossRefGoogle Scholar
  20. Hofert, M. (2010). Sampling nested Archimedean copulas with applications to CDO pricing. PhD thesis, Südwestdeutscher Verlag für Hochschulschriften AG & Co. KG. ISBN 978-3-8381-1656-3.Google Scholar
  21. Hofert, M. (2011). Effciently sampling nested Archimedean copulas. Computational Statistics & Data Analysis, 55, 57–70.MathSciNetCrossRefGoogle Scholar
  22. Hofert, M. (2012a). A stochastic representation and sampling algorithm for nested Archimedean copulas. Journal of Statistical Computation and Simulation, 82(9), 1239–1255.MathSciNetCrossRefGoogle Scholar
  23. Hofert, M. (2012b). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1). https://doi.org/10.1145/2043635.2043638.MathSciNetCrossRefGoogle Scholar
  24. Hofert, M. (2013). On sampling from the multivariate t distribution. The R Journal, 5(2), 129–136.Google Scholar
  25. Hofert, M., Mächler, M., & McNeil, A. J. (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins. Journal of Multivariate Analysis, 110, 133–150.MathSciNetCrossRefGoogle Scholar
  26. Hofert, M., Mächler, M., & McNeil, A. J. (2013). Archimedean copulas in high dimensions: Estimators and numerical challenges motivated by financial applications. Journal de la Société Française de Statistique, 154(1), 25–63.MathSciNetzbMATHGoogle Scholar
  27. Hofert, M., & Pham, D. (2013). Densities of nested Archimedean copulas. Journal of Multivariate Analysis, 118, 37–52.MathSciNetCrossRefGoogle Scholar
  28. Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.CrossRefGoogle Scholar
  29. Kano, Y. (1994). Consistency property of elliptical probability density functions. Journal of Multivariate Analysis, 51, 139–147.MathSciNetCrossRefGoogle Scholar
  30. Khoudraji, A. (1995). Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées. PhD thesis, Québec, Canada: Université Laval.Google Scholar
  31. Kojadinovic, I., & Yan, J. (2010a). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics, 47, 52–63.MathSciNetzbMATHGoogle Scholar
  32. Liebscher, E. (2008). Construction of asymmetric multivariate copulas. Journal of Multivariate Analysis, 99, 2234–2250.MathSciNetCrossRefGoogle Scholar
  33. Malov, S. V. (2001). On finite-dimensional Archimedean copulas. In N. Balakrishnan, I. A. Ibragimov & V. B. Nevzorov (Eds.), Asymptotic methods in probability and statistics with applications (pp. 19–35). Basel: Birkhäuser.CrossRefGoogle Scholar
  34. Marshall, A. W., & Olkin, I. (1988). Families of multivariate distributions. Journal of the American Statistical Association, 83(403), 834–841.MathSciNetCrossRefGoogle Scholar
  35. McNeil, A. J. (2008). Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581.MathSciNetCrossRefGoogle Scholar
  36. McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (2nd ed.). Princeton, NJ: Princeton University Press.zbMATHGoogle Scholar
  37. McNeil, A. J., & Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l 1-norm symmetric distributions. The Annals of Statistics, 37(5b), 3059–3097.MathSciNetCrossRefGoogle Scholar
  38. McNeil, A. J., & Nešlehová, J. (2010). From Archimedean to Liouville copulas. Journal of Multivariate Analysis, 101, 1772–1790.MathSciNetCrossRefGoogle Scholar
  39. Nolan, J. P. (2018). Stable distributions—Models for heavy tailed data. In progress, Chapter 1 online at http://fs2.american.edu/jpnolan/www/stable/stable.html. Boston: Birkhauser.
  40. Pickands, J. (1981). Multivariate extreme value distributions. With a discussion. Bulletin de l’Institut international de statistique, 49, 859–878, 894–902. Proceedings of the 43rd session of the Internatinal Statistical Institute.Google Scholar
  41. Ressel, P. (2013). Homogeneous distributions — And a spectral representation of classical mean values and stable tail dependence functions. Journal of Multivariate Analysis, 117, 246–256.MathSciNetCrossRefGoogle Scholar
  42. Ripley, B. D. (2017). MASS: Support functions and datasets for Venables and Ripley’s MASS. R package version 7.3–47. http://CRAN.R-project.org/package=MASS.Google Scholar
  43. Wang, X., & Yan, J. (2013). Practical notes on multivariate modeling based on elliptical copulas. Journal de la Société Française de Statistique, 154(1), 102–115.MathSciNetzbMATHGoogle Scholar
  44. Yan, J. (2007). Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Marius Hofert
    • 1
  • Ivan Kojadinovic
    • 2
  • Martin Mächler
    • 3
  • Jun Yan
    • 4
  1. 1.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Laboratory of Mathematics and its ApplicationsUniversity of Pau and Pays de l’AdourPauFrance
  3. 3.Seminar for StatisticsETH ZurichZurichSwitzerland
  4. 4.Department of StatisticsUniversity of ConnecticutStorrsUSA

Personalised recommendations