Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Directional $ρ$-coefficients

Enrique de Amo, David García-Fernández, Manuel Úbeda-Flores

TL;DR

The paper extends directional $\rho$-coefficients to arbitrary dimensions, providing a general definition, fundamental properties, and a key representation expressing any direction coefficient as a linear combination of lower-dimensional $\rho^-$ terms. It resolves a conjecture from prior work and corrects aspects of earlier results, grounding the theory in copula-based dependence and orthant concepts. It also develops nonparametric, rank-based estimators with asymptotic guarantees and demonstrates their performance via simulations, plus practical applications to real-world multivariate data. The work advances detection of directional dependencies in high-dimensional settings with potential impact across statistics, finance, and environmental sciences.

Abstract

In this paper we obtain advances for the concept of directional $ρ$-coefficients, originally defined for the trivariate case in [Nelsen, R.B., Úbeda-Flores, M. (2011). Directional dependence in multivariate distributions. Ann. Inst. Stat. Math 64, 677-685] by extending it to encompass arbitrary dimensions and directions in multivariate space. We provide a generalized definition and establish its fundamental properties. Moreover, we resolve a conjecture from the aforementioned work by proving a more general result applicable to any dimension, correcting a result in [García, J.E., González-López, V.A., Nelsen, R.B. (2013). A new index to measure positive dependence in trivariate distributions. J. Multivariate Anal. 115, 481-495] an erratum in the current literature. Our findings contribute to a deeper understanding of multivariate dependence and association, offering novel tools for detecting directional dependencies in high-dimensional settings. Finally, we introduce nonparametric estimators, based on ranks, for estimating directional $ρ$-coefficients from a sample.

Directional $ρ$-coefficients

TL;DR

The paper extends directional -coefficients to arbitrary dimensions, providing a general definition, fundamental properties, and a key representation expressing any direction coefficient as a linear combination of lower-dimensional terms. It resolves a conjecture from prior work and corrects aspects of earlier results, grounding the theory in copula-based dependence and orthant concepts. It also develops nonparametric, rank-based estimators with asymptotic guarantees and demonstrates their performance via simulations, plus practical applications to real-world multivariate data. The work advances detection of directional dependencies in high-dimensional settings with potential impact across statistics, finance, and environmental sciences.

Abstract

In this paper we obtain advances for the concept of directional -coefficients, originally defined for the trivariate case in [Nelsen, R.B., Úbeda-Flores, M. (2011). Directional dependence in multivariate distributions. Ann. Inst. Stat. Math 64, 677-685] by extending it to encompass arbitrary dimensions and directions in multivariate space. We provide a generalized definition and establish its fundamental properties. Moreover, we resolve a conjecture from the aforementioned work by proving a more general result applicable to any dimension, correcting a result in [García, J.E., González-López, V.A., Nelsen, R.B. (2013). A new index to measure positive dependence in trivariate distributions. J. Multivariate Anal. 115, 481-495] an erratum in the current literature. Our findings contribute to a deeper understanding of multivariate dependence and association, offering novel tools for detecting directional dependencies in high-dimensional settings. Finally, we introduce nonparametric estimators, based on ranks, for estimating directional -coefficients from a sample.

Paper Structure

This paper contains 13 sections, 9 theorems, 77 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Copulas and Sklar's theorem
  4. Measures of association
  5. Directional dependence
  6. Multivariate directional $\rho$-coefficients
  7. Nonparametric estimators for directional dependence
  8. Empirical processes and some properties of these estimators
  9. Empirical process
  10. Properties of the estimators
  11. Simulations
  12. Applications
  13. Conclusion

Key Result

Theorem 1

Let ${\mathbf X}=(X_1,X_2,...,X_n)$ be a random $n$-vector with joint distribution function $H$ and one-dimensional marginal distributions $F_1,F_2,...,F_n$. Then there exists an $n$-copula (uniquely determined on $\times_{i=1}^n\text{ Range } (F_i)$) such that for all ${\mathbf x}\in[-\infty,+\infty]^n$. If all the marginals $F_i$ are continuous, then the $n$-copula is unique.

Figures (1)

  • Figure 1: Scatterplots of log-return data.

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1: Sklar
  • Definition 2: PD$(\alpha)$ dependence concept
  • Definition 3: PD$(\alpha)$ order
  • Definition 4
  • Corollary 2
  • proof
  • Example 1: Directional $\rho$-coefficients for $\Pi_n$
  • Example 2: Directional $\rho$-coefficients for $M_n$
  • Example 3: Directional $\rho$-coefficients for the Farlie-Gumbel-Morgenstern (FGM) family of $n$-copulas
  • ...and 15 more