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

TL;DR

This paper introduces an entropy-based measure to quantify upper quantile interdependence in multivariate distributions, invariant to marginal transformations, with applications to financial data and extreme value theory.

Contribution

It proposes a novel entropy index for measuring interdependence along the main diagonal of the copula, linking it to extremal coefficients in extreme value theory.

Findings

The entropy index effectively quantifies interdependence in multivariate data.

Application to stock prices demonstrates practical utility.

The index converges to the extremal coefficient for extreme value distributions.

Abstract

We introduce a new measure of interdependence among the components of a random vector along the main diagonal of the vector copula, i.e. along the line $u_{1}=\ldots=u_{J}$, for $\left(u_{1},\ldots,u_{J}\right)\in\left[0,1\right]^{J}$. Our measure is related to the Shannon entropy of a discrete random variable, hence we call it an "entropy index". This entropy index is invariant with respect to marginal non-decreasing transformations and can be used to quantify the intensity of the vector components association 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 is in the domain of attraction of an extreme value distribution, our index is shown to have as limit the distribution's extremal coefficient, which can be interpreted as the effective number of asymptotically independent components in the vector.