On a strong metric on the space of copulas and its induced dependence measure

Abstract

Using the one-to-one correspondence between copulas and Markov operators on L1([0,1]) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D1 on the space of copulas C that is a metrization of the strong operator topology of the corresponding Markov operators. It is shown that the resulting metric space (C,D1) is complete and separable and that the induced dependence measure Ζ1, defined as a scalar times the D1-distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D1-distance to the product copula is exactly the class of completely dependent copulas, i.e. copulas induced by Lebesgue-measure preserving transformations on [0,1]. Hence, in contrast to the uniform distance d∞, Π cannot be approximated arbitrarily well by completely dependent copulas with respect to D1. The interrelation between D1 and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between Ζ1 and the mutual dependence measure Ω by Siburg and Stoimenov is analyzed. Ζ1 is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given. © 2011 Elsevier Inc.