An information theoretic approach to statistical dependence: copula information

TL;DR

This paper explores the use of copula functions to decompose multivariate information into marginal and dependence parts, introducing measures like information excess and demonstrating their application in financial data analysis.

Contribution

It introduces an information-theoretic framework for analyzing copulas, including new measures like information excess and analytical expressions for T-copulas.

Findings

Mutual information bounds empirical log-likelihood of copulas.

Information excess quantifies deviation from maximum entropy.

Framework applied successfully to financial data.

Abstract

We discuss the connection between information and copula theories by showing that a copula can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. We define the information excess as a measure of deviation from a maximum entropy distribution. The idea of marginal invariant dependence measures is also discussed and used to show that empirical linear correlation underestimates the amplitude of the actual correlation in the case of non-Gaussian marginals. The mutual information is shown to provide an upper bound for the asymptotic empirical log-likelihood of a copula. An analytical expression for the information excess of T-copulas is provided, allowing for simple model identification within this family. We illustrate the framework in a financial data set.