Entropy measure for the quantification of upper quantile interdependence in multivariate distributions

We study the applicability of a measure of interdependence among the components of a random vector along the main diagonal of the vector’s copula, i.e. along the line u1=⋯=uJu1=⋯=uJ, for (u1,…,uJ)∈[0,1]J(u1,…,uJ)∈[0,1]J. Our measure is related to the Shannon entropy of a discrete random variable, and it is a measure of local divergence between the data empirical copula and the independent copula. Hence we call it a “local dependence entropy index”, which can be interpreted as an effective number of independent variables. It is invariant with respect to marginal non-decreasing transformations and can be used in arbitrary dimensions. We show the applicability of our entropy index by an example with real data of 4 stock prices of the DAX index. In case the random vector possesses an extreme value copula, the index is shown to have as limit the extremal coefficient, which can be interpreted as the effective number of asymptotically independent components in the vector.