Discussion to the paper ‘Robust Distance Covariance’

Abstract

We would like to thank S. Leyder, J. Raymaekers, and P. Rousseeuw for their insightful andwell-written article ‘Robust Distance Covariance’. The paper provides a clear and comprehen-sive overview of distance covariance and related concepts and derives their influence functionsand breakdown points. It is shown that, depending on parameter choices, classical distance co-variance can exhibit a degree of robustness, although it still falls somewhat short in this regard.To address this limitation, the authors introduce a novel robust version of distance covari-ance. By transforming the data into a higher-dimensional bounded domain, their methodachieves a high breakdown point while preserving the desirable property that a zero value ofdistance covariance implies independence between the random elements involved.Historically, the robust statistics literature has paid limited attention to the properties of sta-tistical methods in settings where independence is the central concept. This may be attributedto the dominance of elliptical models, where independence is typically not a relevant notion.However, the growing popularity of models such as the independent component model (seeNordhausen and Oja 2018) has increasingly motivated the investigation of robust estimationtechniques in scenarios where independence plays a critical role (Nordhausen and Tyler 2015).These investigations have revealed that many scatter functionals fail to return zeros at theappropriate entries when components are independent, especially in the presence ofasymmetry. As shown in Oja et al. (2006) and Nordhausen and Tyler (2015), a fruitful strategyis to symmetrise the data and operate on pairwise differences—a principle that is alsoinherent in the concept of distance covariance, as further highlighted in Raymaekers andRousseeuw (2025).In this discussion, we are particularly interested in the application of (robust) generalizeddistance covariance in multivariate contexts, extending beyond the primarily bivariate focusof S. Leyder, J. Raymaekers, and P. Rousseeuw’s article. At first glance, it may seem unusualthat, regardless of the dimensionality of the vectors considered, (robust) generalized distancecovariance returns a scalar rather than a matrix. From a multivariate perspective, this naturallyinvites consideration of the method within the framework of projection pursuit.