First digit distributions of partial derivatives of copulas

Abstract

We propose an indicator of dependence between two random variables X and Y whose distribution functions are continuous. The starting point of our investigation is the well-known fact that if C is the copula of X and Y , then the conditional expectations of Y given X and of X given Y are completely determined by the first partial derivatives of C. We then investigate whether the distributions of the first digits of @1C and @2C can give rise to a measure of dependence. Our candidates are normalized maxima of the first digit distributions of @iC. We also show that these normalized maxima attain minimum value if X and Y are independent and attain maximum value when they are completely dependent. However, both converses are not true as we are able to produce counterexamples. At last, we introduce a numerical computation of the dependence indicators. Although neither independence nor complete dependence can be characterized by the dependence indicators, we illustrate via some parametric classes of copulas that their numerical values are directly proportional to those of the dependence measures introduced by Siburg-Stoimenov and of Trutschnig.