Discussion of ‘Robust distance covariance’ by S. Leyder, J. Raymaekers and P. J. Rousseeuw

Abstract

The distance covariance in Székely et al. (2007) was introduced for $\left({ℝ}^{d_1}\times {ℝ}^{d_2}\right)$-valued data. Building on the concepts introduced in Chernozhukov et al. (2017) and Hallin et al. (2021), the corresponding rank-based versions only came some 10 years later, with Shi et al. (2022a, 2022b, 2024) and, in a slightly different form, with Deb and Sen (2023). We only briefly sketch their definitions (Section 2.1) and suggest (Section 2.2) how to robustify them, along the same lines as in [LRR], into statistics with redescending influence functions.

Section 2 takes care of distance covariances for multivariate real data. Further extensions, however, are highly desirable and have not been explored so far. In this section, we are sketching a possible extension to directional data, that is, to random directions distributed on hyperspheres. We conjecture that extensions to other compact nonlinear Riemannian manifolds, such as tori and products of hyperspheres, are feasible using the same methodology, along with the nonparametric tools for Riemannian manifolds developed by Hallin and Liu (2024). Here, we restrict the exposition to the case of hyperspheres.

Building on the measure-transportation-based notions of directional ranks and signs defined in Hallin et al. (2024), we show how the ‘robustness via transformation’ principle emphasized by [LRR] extends beyond the case of bivariate independence and also applies in higher-dimensional Euclidean spaces and on compact manifolds. The case of directional variables (taking values on (hyper)spheres) is given special attention: although (hyper)spheres are not metric spaces of the strong negative type, we introduce a class of pertinent directional distance covariance test statistics and illustrate their applicability. We hope that this note will open the door to further research directions in line with [LRR].