Abstract
This chapter introduces the main copula classes and the corresponding sampling procedures, along with some copula transformations that are important for practical purposes.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Beirlant, J., Goegebeur, Y., Segers, J., & Teugels, J. (2004). Statistics of extremes: Theory and applications. Wiley series in probability and statistics. Chichester: Wiley.
Bolker, B. (2017). bbmle: Tools for general maximum likelihood estimation. R package version 1.0.20. https://CRAN.R-project.org/package=bbmle.
Capéraà, P., Fougères, A.-L., & Genest, C. (2000). Bivariate distributions with given extreme value attractor. Journal of Multivariate Analysis, 72, 30–49.
Charpentier, A., Fougères, A.-L., Genest, C., & Nešlehová, J. G. (2014). Multivariate Archimax copulas. Journal of Multivariate Analysis, 126, 118–136.
De Haan, L., & Ferreira, A. (2006). Extreme value theory: An introduction. New York: Springer.
Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review, 73(1), 111–129.
Devroye, L. (1993). A triptych of discrete distributions related to the stable law. Statistics & Probability Letters, 18(5), 349–351.
Fang, K.-T., Kotz, S., & Ng, K.-W. (1990). Symmetric multivariate and related distributions. London: Chapman & Hall.
Feller, W. (1971). An introduction to probability theory and its applications (2nd ed.), vol. 2. New York: Wiley.
Galambos, J. (1978). The asymptotic theory of extreme order statistics. Wiley series in probability and mathematical statistics. New York: Wiley.
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.
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.
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.
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.
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.
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.
Gudendorf, G., & Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. Journal of Statistical Planning and Inference, 143, 3073–3085.
Gumbel, E. J. (1958). Statistics of extremes. New York: Columbia University Press.
Hofert, M. (2008). Sampling Archimedean copulas. Computational Statistics & Data Analysis, 52, 5163–5174.
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.
Hofert, M. (2011). Effciently sampling nested Archimedean copulas. Computational Statistics & Data Analysis, 55, 57–70.
Hofert, M. (2012a). A stochastic representation and sampling algorithm for nested Archimedean copulas. Journal of Statistical Computation and Simulation, 82(9), 1239–1255.
Hofert, M. (2012b). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1). https://doi.org/10.1145/2043635.2043638.
Hofert, M. (2013). On sampling from the multivariate t distribution. The R Journal, 5(2), 129–136.
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.
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.
Hofert, M., & Pham, D. (2013). Densities of nested Archimedean copulas. Journal of Multivariate Analysis, 118, 37–52.
Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.
Kano, Y. (1994). Consistency property of elliptical probability density functions. Journal of Multivariate Analysis, 51, 139–147.
Khoudraji, A. (1995). Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées. PhD thesis, Québec, Canada: Université Laval.
Kojadinovic, I., & Yan, J. (2010a). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics, 47, 52–63.
Liebscher, E. (2008). Construction of asymmetric multivariate copulas. Journal of Multivariate Analysis, 99, 2234–2250.
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.
Marshall, A. W., & Olkin, I. (1988). Families of multivariate distributions. Journal of the American Statistical Association, 83(403), 834–841.
McNeil, A. J. (2008). Sampling nested Archimedean copulas. Journal of Statistical Computation and Simulation, 78(6), 567–581.
McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (2nd ed.). Princeton, NJ: Princeton University Press.
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.
McNeil, A. J., & Nešlehová, J. (2010). From Archimedean to Liouville copulas. Journal of Multivariate Analysis, 101, 1772–1790.
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.
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.
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.
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.
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.
Yan, J. (2007). Enjoy the joy of copulas: With a package copula. Journal of Statistical Software, 21(4), 1–21.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Hofert, M., Kojadinovic, I., Mächler, M., Yan, J. (2018). Classes and Families. In: Elements of Copula Modeling with R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-89635-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-89635-9_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-89634-2
Online ISBN: 978-3-319-89635-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)