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arXiv · 2505.22206

Directional $\rho$-coefficients

Enrique de Amo·David García-Fernández·Manuel Úbeda-Flores
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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.

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Enrique de Amo, David García-Fernández, Manuel Úbeda-Flores. 2025-05-28. Directional $\rho$-coefficients. https://arxiv.org/abs/2505.22206

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