Directional $\rho$-coefficients

TL;DR

This paper extends the concept of directional $ ho$-coefficients to arbitrary dimensions, proves a conjecture, corrects previous results, and introduces rank-based estimators for high-dimensional dependence analysis.

Contribution

It generalizes directional $ ho$-coefficients to any dimension, proves a conjecture, corrects prior results, and develops nonparametric estimators for multivariate dependence.

Findings

Generalized directional $ ho$-coefficients to arbitrary dimensions.

Proved a conjecture and corrected previous results.

Introduced rank-based estimators for practical application.

Abstract

In this paper we obtain advances for the concept of directional $\rho$-coefficients, originally defined for the trivariate case in [Nelsen, R.B., \'Ubeda-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 [Garc\'ia, J.E., Gonz\'alez-L\'opez, 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 $\rho$-coefficients from a sample.