Multivariate Information Measures: A Copula-based Approach

TL;DR

This paper introduces a novel copula-based information measure for multivariate data, exploring its properties, estimators, and applications in dependency modeling and goodness-of-fit testing.

Contribution

It presents a new multivariate cumulative copula entropy, its properties, estimators, and a copula distance measure, advancing dependency quantification in multivariate datasets.

Findings

Defined a multivariate cumulative copula entropy and its properties.

Developed a non-parametric estimator using empirical beta copula.

Proposed a copula distance measure based on Kullback-Leibler divergence.

Abstract

Multivariate datasets are common in various real-world applications. Recently, copulas have received significant attention for modeling dependencies among random variables. A copula-based information measure is required to quantify the uncertainty inherent in these dependencies. This paper introduces a multivariate variant of the cumulative copula entropy and explores its various properties, including bounds, stochastic orders, and convergence-related results. Additionally, we define a cumulative copula information generating function and derive it for several well-known families of multivariate copulas. A fractional generalization of the multivariate cumulative copula entropy is also introduced and examined. We present a non-parametric estimator of the cumulative copula entropy using empirical beta copula. Furthermore, we propose a new distance measure between two copulas based on the Kullback-Leibler divergence and discuss a goodness-of-fit test based on this measure.