Abstract
In Lauterbach (ZVersWiss, 101(5), 605–619, 2012) and Lauterbach and Pfeifer (Copulae in mathematical and quantitative finance, Springer, Dordrecht, 2013) the family of Singular Mixture Copulas was introduced. We present and discuss two extensions of these copulas. Both extensions are based on an approach introduced by Khoudraji (Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées. Ph.D. thesis, 1995). We study the dependence properties of the constructed copulas and show that the resulting copulas possess differing upper and lower tail dependence coefficients.
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References
Cherubini, U., Luciano, E., & Vecchiato, W. (2004). Copula methods in finance. Chichester: Wiley.
Deheuvels, P. (1978). Caractérisation complète des lois extrêmes multivariées et de la convergence des types extrêmes. Publication Institute Statistics University of Paris (Vol. 23, pp.1–37).
Dovgoshey, O., Martio, O., Ryazanov, V., & Vuorinen, M. (2006). The cantor function. Expositiones Mathematicae, 24, 1–37.
Durante, F. (2009). Construction of non-exchangeable bivariate distribution functions. Statistical Papers, 50, 383–391.
Durante, F., & Sempi, C. (2010). Copula theory: An introduction. In P. Jaworski, F. Durante, W. Härdle, & T. Rychlik (Eds.), Copula theory and its applications (pp. 3–31). Dordrecht: Springer.
Embrechts, P., Lindskog, F., & McNeil, A. J. (2003). Modelling dependence with copulas and applications to risk management. In S. T. Rachev (Ed.), Handbook of heavy tailed distributions in finance (pp. 329–384). Amsterdam: Elsevier.
Fréchet, M.: Sur les tableaux de corrélation dont les marges sont données. Annales de l’Université de Lyon. Section A, 9, 53–77 (1951).
Genest, C., Ghoudi, K., & Rivest, L.-P. (1998). Understanding relationships using copulas, by Edward Frees and Emiliano Valdez, January 1998. North American Actuarial Journal, 2(3), 143–149.
Genest, C., & MacKay, R. J. (1986). The joy of copulas: Bivariate distributions with uniform marginals. The American Statistician, 40(4), 280–283.
Hoeffding, W. (1940). Maßstabinvariante Korrelationstheorie. Schriften des Mathematischen Instituts und des Instituts für Angewandte Mathematik der Universität Berlin, 5, 179–233.
Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.
Khoudraji, A. (1995). Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées. Ph.D. thesis, Université de Laval, Québec (Canada).
Lauterbach, D. (2012). Singular mixture copulas: A geometric method of constructing copulas. ZVersWiss, 101(5), 605–619.
Lauterbach, D., & Pfeifer, D. (2013). Singular mixture copulas. In P. Jaworski, F. Durante, & W. Härdle (Eds.), Copulae in mathematical and quantitative finance (pp. 165–175). Dordrecht: Springer.
McNeil, A. J., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts techniques and tools. Princeton: Princeton University Press.
Nelsen, R. B. (2006). An introduction to Copulas. New York: Springer.
Sklar, A. (1959). Fonctions de répartition à \(n\) dimensions et leurs marges. Publication Institute Statistics University of Paris (Vol. 8, pp. 229–231).
Wolff, E.F. (1977). Measures of dependence derived from copulas. Ph.D. thesis, University of Massachusetts, Amherst.
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The authors would like to sincerely thank the reviewers for their detailed comments, which led to an improvement of the presentation of this paper.
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Lauterbach, D., Pfeifer, D. (2015). Some Extensions of Singular Mixture Copulas. In: Hallin, M., Mason, D., Pfeifer, D., Steinebach, J. (eds) Mathematical Statistics and Limit Theorems. Springer, Cham. https://doi.org/10.1007/978-3-319-12442-1_15
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DOI: https://doi.org/10.1007/978-3-319-12442-1_15
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