Conference Agenda

Measures of association based on ranks, such as Spearman’s footrule[1], play a central role in multivariate statistics due to their robustness and invariance properties. However, classical versions of these coefficients are often unable to capture directional dependence structures that arise in high-dimensional settings. Motivated by this limitation and by the newly defined coefficients described subsequently[2] [3], we introduce a novel family of directional Spearman’s footrule coefficients designed to quantify multivariate dependence along prescribed directions in the unit d-dimensional hypercube.

The proposed coefficients are formulated within the framework of copula theory, which allows for a clear separation between marginal behavior and the underlying dependence structure. Our construction extends the classical Spearman’s footrule by incorporating directional information, enabling the detection of dependence patterns that remain undetected by standard measures. We establish a general definition for arbitrary dimensions and directions and investigate its main theoretical properties. In particular, we analyze their behavior under independence and maximal positive dependence, their relation to stochastic orders, as well as their relationship with marginal distributions and lower-dimensional structures. These properties are shown to be consistent with those of the classical footrule coefficient.

To facilitate practical implementation, we also introduce nonparametric estimators based on ranks. These estimators are easy to compute and suitable for multivariate data. Their asymptotic behavior is discussed, highlighting consistency and stability properties analogous to those of existing rank-based dependence measures.

Several illustrative examples are provided to demonstrate the usefulness of the proposed coefficients. Explicit expressions are derived for well-known families of d-copulas, including the Farlie–Gumbel–Morgenstern and Cuadras–Augé, allowing for a detailed analysis of how directional dependence varies with model parameters. These examples show that the proposed coefficients are able to distinguish different directional dependence patterns even when classical global measures coincide.

Overall, this work provides a new tool for directional dependence analysis in multivariate settings, complementing existing rank-based measures and offering a finer understanding of complex dependence structures with applications in finance, reliability, and multivariate risk analysis.

[1] Spearman, C. (1906). ‘Footrule’ for measuring correlation. Brithis Journal of Psychology, 2, 89-108.

[2] Úbeda-Flores, M. (2004). Multivariate versions of Blomqvist’s beta and Spearman’s footrule. Ann. Inst. Statist. Math., 57(4), 781-788.

[3] Decancq, K., Pérez, A., Prieto-Alaiz, M. (2025). Multivariate Dependence Based on Diagonal Sections: Spearman’s Footrule and Related Measures. In: Steland, A., Rafajłowicz, E., Parolya, N. (eds) Stochastic Models, Statistics and Their Applications. SMSA 2024. Springer Proceedings in Mathematics & Statistics, vol 499. Springer, Cham.