For continuous random variables, many dependence concepts and measures of association can be expressed in terms of the corresponding copula only and are thus independent of the marginal distributions. This interrelationship generally fails as soon as there are discontinuities in the marginal distribution functions. In this paper, we consider an alternative transformation of an arbitrary random variable to a uniformly distributed one. Using this technique, the class of all possible copulas in the general case is investigated. In particular, we show that one of its members-the standard extension copula introduced by Schweizer and Sklar-captures the dependence structures in an analogous way the unique copula does in the continuous case. Furthermore, we consider measures of concordance between arbitrary random variables and obtain generalizations of Kendall's tau and Spearman's rho that correspond to the sample version of these quantities for empirical distributions.
|Seiten (von - bis)||544-567|
|Fachzeitschrift||Journal of Multivariate Analysis|
|Publikationsstatus||Veröffentlicht - März 2007|
Bibliographische NotizFunding Information:
This work was partly supported by RiskLab, ETH Zürich. The author is grateful to Professors Dietmar Pfeifer and Paul Embrechts for many fruitful discussions and support. She would also like to thank Prof. Marco Scarsini and Dr. Alfred Müller as well as the referees for several useful comments.