Csisz\'ar indices and interpolating copulas

Abstract

We study various properties of $f$-divergences and Csisz\'ar indices between two probability distributions in very general setups for the convex function $f$ and for the probability distributions. We establish general structural properties of $f$-divergences and show how they are inherited by the associated Csisz\'ar indices, including monotonicity and invariance under suitable transformations. We also study the relationship between Csisz\'ar indices and copula representations of random vectors. When the marginal distributions have atoms, the copula representation is not unique and the Csisz\'ar index of the transformed vectors may increase. We build a large family of interpolating copulas which minimize the Csisz\'ar index and thus preserve the dependence structure of the initial vector.