Kybernetika 52 no. 5, 785-790, 2016

Proving the characetrization of Archimedean copulas via Dini derivatives

Juan Fernández-Sánchez and Manuel Úbeda-FloresDOI: 10.14736/kyb-2016-5-0785

Abstract:

In this note we prove the characterization of the class of Archimedean copulas by using Dini derivatives.

Keywords:

Archimedean copula, Dini derivative, derived number

Classification:

60E05, 62E10

References:

  1. C. Alsina, M. J. Frank and B. Schweizer: Associative Functions: Triangular Norms and Copulas. World Scientific, Singapore 2006.   DOI:10.1142/9789812774200
  2. E. de Amo, M. Díaz Carrillo and J. Fernández Sánchez: Characterization of all copulas associated with non-continuous random variables. Fuzzy Sets Syst. 191 (2012), 103-112.   DOI:10.1016/j.fss.2011.10.005
  3. L. Berg and M. Krüppel: De Rahm's singular function and related functions. Z. Anal. Anw. 19 (2000), 227-237.   DOI:10.4171/zaa/947
  4. U. Cherubini, E. Luciano and W. Vecchiato: Copula Methods in Finance. Wiley Finance Series, John Wiley and Sons Ltd., Chichester 2004.   DOI:10.1002/9781118673331
  5. F. Durante and P. Jaworski: A new characterization of bivariate copulas. Comm. Statist. Theory Methods 39 (2010), 2901-2912.   DOI:10.1080/03610920903151459
  6. F. Durante and C. Sempi: Principles of Copula Theory. Chapman and Hall/CRC, London 2015.   DOI:10.1201/b18674
  7. C. Genest and J. MacKay: Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 (1986), 145-159.   DOI:10.2307/3314660
  8. J. W. Hagood and B. S. Thomson: Recovering a function from a Dini derivative. Amer. Math. Monthly 113 (2006), 34-46.   DOI:10.2307/27641835
  9. P. Jaworski, F. Durante, W. Härdle and T. Rychlik (editors): Copula Theory and its Applications. Lecture Notes in Statistics-Proceedings, Springer, Berlin-Heidelberg 2010.   DOI:10.1007/978-3-642-12465-5
  10. C. H. Ling: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189-212.   CrossRef
  11. S. Łojasiewicz: An Introduction to the Theory of Real Functions. Third Edition. A Wiley-Interscience Publication, John Wiley and Sons Ltd., Chichester 1988.   CrossRef
  12. A. J. McNeil and J. Nešlehová: Multivariate Archimedean copulas, $d$-monotone functions and $l_1$-norm symmetric distributions. Ann. Stat. 37 (2009), 3059-3097.   DOI:10.1214/07-aos556
  13. A. J. McNeil, R. Frey and P. Embrechts: Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton 2005.   CrossRef
  14. L. P. Natanson: Theory of Functions of a Real Variable. Vol. I, revised edition. Frederick Ungar Publishing, New York 1961.   CrossRef
  15. R. B. Nelsen: An Introduction to Copulas. Second Edition. Springer, New York 2006.   DOI:10.1007/0-387-28678-0
  16. B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North-Holland, New York 1983. Reprinted, Dover, Mineola NY, 2005.   CrossRef
  17. A. Sklar: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229-231.   CrossRef
  18. W. Wysocki: Characterizations of Archimedean n-copulas. Kybernetika 51 (2015), 212-230.   DOI:10.14736/kyb-2015-2-0212