On concordance measures and copulas with fractal support
نویسندگان
چکیده
Copulas can be used to describe multivariate dependence structures. We explore the rôle of copulas with fractal support in the study of association measures. 1 General introduction and motivation Copulas are of interest because they link joint distributions to their marginal distributions. Sklar [12] showed that, for any real-valued random variables X1 and X2 with joint distribution H, there exists a copula C such that H(u, v) = C(F1(u), F2(v)), where F1 and F2 denote the cumulative (or margin) distributions of X1 and X2, respectively. If the marginals are continuous, then the copula is unique. Notice that it is also true the converse implication of Sklar’s Theorem. In fact, we may link any univariate distributions with any copula in order to obtain a valid joint distribution function. An implication of Sklar Theorem is that the dependence among X1 and X2 is fully described by the associated copula. Indeed, most conventional dependence measures can be explicitly expressed in terms of the copula, and they are designed to capture certain aspects of dependence or association between random variables. On the other hand, all the examples of singular copulas we have found in the literature are supported by sets with Hausdorff dimension 1. However, it is implicit in some papers, for example in [11], that the well known examples of Peano and Hilbert curves provide self-similar copulas with fractal support, since the Hausdorff dimension of their graphs is 3/2. Departamento de Álgebra y Análisis Matemático. Universidad of Almeŕıa, Spain. [email protected] · Departamento de Análisis Matemático. Universidad of Granada, Spain [email protected]
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