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 L 1 ( [ 0 , 1 ] ) and expressing the Markov operators in terms of regular conditional distributions (Markov kernels) allows to define a metric D 1 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 , D 1 ) is complete and separable and that the induced dependence measure ζ 1 , defined as a scalar times the D 1 -distance to the product copula Π, has various good properties. In particular the class of copulas that have maximum D 1 -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 D 1 . The interrelation between D 1 and the so-called ∂-convergence by Mikusinski and Taylor as well as the interrelation between ζ 1 and the mutual dependence measure ω by Siburg and Stoimenov is analyzed. ζ 1 is calculated for some well-known parametric families of copulas and an application to singular copulas induced by certain Iterated Functions Systems is given.