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Multivariate Information Measures: A Copula-based Approach

Mohd. Arshad, Swaroop Georgy Zachariah, Ashok Kumar Pathak

TL;DR

This work develops a copula-based information-theoretic framework for multivariate dependence by introducing the multivariate cumulative copula entropy $\zeta(C)$, its information generating function $\mathcal{G}_C(s)$, and a fractional extension $\zeta_r(C)$. It provides closed-form analyses for key copulas, establishes bounds, ordering properties, and convergence results, and extends to non-parametric estimation via the empirical beta copula with guaranteed consistency. A KL-based copula distance $CCKL(C_1:C_2)$ is proposed and used to construct a robust goodness-of-fit test for copulas, including a practical Monte Carlo-based procedure. The methodology is validated through simulations and applied to the Pima Indians Diabetes data, where the Frank copula is identified as the best fit by the proposed distance, highlighting the framework’s practical utility for copula selection and dependence quantification in multivariate settings.

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.

Multivariate Information Measures: A Copula-based Approach

TL;DR

This work develops a copula-based information-theoretic framework for multivariate dependence by introducing the multivariate cumulative copula entropy , its information generating function , and a fractional extension . It provides closed-form analyses for key copulas, establishes bounds, ordering properties, and convergence results, and extends to non-parametric estimation via the empirical beta copula with guaranteed consistency. A KL-based copula distance is proposed and used to construct a robust goodness-of-fit test for copulas, including a practical Monte Carlo-based procedure. The methodology is validated through simulations and applied to the Pima Indians Diabetes data, where the Frank copula is identified as the best fit by the proposed distance, highlighting the framework’s practical utility for copula selection and dependence quantification in multivariate settings.

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.
Paper Structure (10 sections, 17 theorems, 77 equations, 4 figures, 10 tables)

This paper contains 10 sections, 17 theorems, 77 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Multivariate Cumulative Copula Entropy
  3. Cumulative Copula Information Generating Function
  4. Fractional Multivariate Cumulative Copula Entropy
  5. Empirical Beta Cumulative Copula Entropy
  6. Cumulative Copula Kullback-Leibler Divergence and its Application
  7. A Goodness of fit test for copula
  8. Simulation Study and Data Analysis
  9. Selection of an appropriate copula for "Pima Indians Diabetes" data
  10. Conclusion

Key Result

Theorem 2.1

For every $k$-dimensional copula, where $\mathcal{B}_k(C)=2^{-k}\left[\dfrac{\rho_k^{-}(C)}{n(k)}+1\right]$ and $\rho_k^{-}(C)$ is the multivariate version of Spearman's correlation defined in Eq. (spm1).

Figures (4)

  • Figure 1: The fractional empirical beta CCE and theoretical fractional CCE of various bivariate copulas.
  • Figure 2: The fractional empirical beta CCE and theoretical fractional CCE of various trivariate copulas.
  • Figure 3: The empirical beta and theoretical CCIGF of various bivariate copulas.
  • Figure 4: The empirical beta and theoretical CCIGF of various trivariate copulas.

Theorems & Definitions (54)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.1
  • proof
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 44 more