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

Testing the null hypothesis ℋ 0 $$ {\mathrm{\mathscr{H}}}_0 $$ of independence between two variables X $$ X $$ and Y $$ Y $$ is an important and fundamental problem in theoretical as well as in applied statistics. Thanks to the fact that they vanish if and only if X $$ X $$ and Y $$ Y $$ are independent, the distance covariance dCov X , Y $$ \left(X,Y\right) $$ and the corresponding distance correlation dCor X , Y $$ \left(X,Y\right) $$ are natural measures of the dependence between X $$ X $$ and Y $$ Y $$ . Their empirical versions constitute tests statistics for testing ℋ 0 $$ {\mathrm{\mathscr{H}}}_0 $$ against the alternative ℋ 1 $$ {\mathrm{\mathscr{H}}}_1 $$ of unspecified forms of dependence. Introduced by Székely et al. (2007), distance covariance and correlation have been thoroughly studied and are used in a number of applications. In the sequel, we concentrate on distance covariance, but distance correlation clearly enjoys similar properties. Point (iii) above highlights what the authors call ‘robustness by transformation’, that is, the fact that, for any bijective transformations ψ 1 $$ {\psi}_1 $$ and ψ 2 $$ {\psi}_2 $$ , X $$ X $$ and Y $$ Y $$ are independent if and only if ψ 1 X $$ {\psi}_1(X) $$ and ψ 2 Y $$ {\psi}_2(Y) $$ are, so that dCov X , Y = 0 $$ \left(X,Y\right)=0 $$ if and only if dCov ψ 1 X , ψ 2 Y = 0 $$ \left({\psi}_1(X),{\psi}_2(Y)\right)=0 $$ . This is an indication that a non-robust characterization of independence can be turned into a robust one via adequate transformations, and a simple but quite valuable finding of the authors is that these transformations ψ 1 $$ {\psi}_1 $$ and ψ 2 $$ {\psi}_2 $$ need not be from ℝ $$ \mathrm{\mathbb{R}} $$ to ℝ $$ \mathrm{\mathbb{R}} $$ . Transformations, moreover, also can be used to avoid the moment assumptions required when considering dCov X , Y $$ \left(X,Y\right) $$ . This is achieved, for instance, by the classical probability integral transformations ψ 1 = F X $$ {\psi}_1={F}_X $$ and ψ 2 = F Y $$ {\psi}_2={F}_Y $$ (where F X $$ {F}_X $$ and F Y $$ {F}_Y $$ stand for the distribution functions of X $$ X $$ and Y $$ Y $$ , respectively), the empirical versions F X n $$ {F}_X^{(n)} $$ and F Y n $$ {F}_Y^{(n)} $$ of which yield the ranks of the observations X 1 , … , X n $$ {X}_1,\dots, {X}_n $$ and Y 1 , … , Y n $$ {Y}_1,\dots, {Y}_n $$ . The resulting dCov F X n X F Y n Y $$ \left({F}_X^{(n)}(X),{F}_Y^{(n)}(Y)\right) $$ then achieves, under the null hypothesis of independence, distribution-freeness (for absolutely continuous X $$ X $$ and Y $$ Y $$ ). That transformation can be combined with the use of scores J 1 $$ {J}_1 $$ and J 2 $$ {J}_2 $$ mapping [0,1) to the real line, yielding rank-based test statistics dCov J 1 ∘ F X n X , J 2 ∘ F Y n Y $$ \left({J}_1\circ {F}_X^{(n)}(X),{J}_2\circ {F}_Y^{(n)}(Y)\right) $$ . The case of Gaussian scores J 1 = Φ − 1 = J 2 $$ {J}_1={\Phi}^{-1}={J}_2 $$ (where Φ $$ \Phi $$ , as usual, denotes the standard normal distribution function), which is briefly mentioned by [LRR] in their sect. 4, corresponds to the traditional (van der Waerden or) Gaussian-score rank statistics; but other scores are possible, such as the Wilcoxon scores ( J 1 u = u = J 2 u $$ {J}_1(u)=u={J}_2(u) $$ ) or the sign test scores ( J 1 u = sign u − 1 / 2 = J 2 u $$ {J}_1(u)=\operatorname{sign}\left(u-1/2\right)={J}_2(u) $$ ). Now, as usual, there is a trade-off between robustness and efficiency: depending on the choice of the ψ 1 $$ {\psi}_1 $$ and ψ 2 $$ {\psi}_2 $$ transformations and the scores J 1 $$ {J}_1 $$ and J 2 $$ {J}_2 $$ , what are, in terms of efficiency, the costs (the benefits) of transformation-robustifying the ‘plain version’ dCov X n Y n $$ \left({X}^{(n)},{Y}^{(n)}\right) $$ of the empirical distance covariance into dCov ψ 1 X n , ψ 2 Y n $$ \left({\psi}_1\left({X}^{(n)}\right),{\psi}_2\left({Y}^{(n)}\right)\right) $$ or dCov J 1 ∘ F X n X , J 2 ∘ F Y n Y ) $$ \left({J}_1\circ {F}_X^{(n)}(X)\right),{J}_2\circ {F}_Y^{(n)}(Y)\Big) $$ ? The answer is not immediate, and that issue, in [LRR], is not considered. Some power features of the rank-based transformations, such as Chernoff–Savage or Hodges–Lehmann results, could be obtained along the lines developed in Hallin and Paindaveine (2008) for the traditional Wilks test of independence. While the introduction, by Székely et al. (2007), of distance covariance was aiming at characterizing independence between vectors (i.e., X , Y $$ \left(\mathbf{X},\mathbf{Y}\right) $$ taking values in ℝ d 1 × ℝ d 2 $$ {\mathrm{\mathbb{R}}}^{d_1}\times {\mathrm{\mathbb{R}}}^{d_2} $$ with max d 1 d 2 > 1 $$ \max \left({d}_1,{d}_2\right)>1 $$ ), [LRR] are limiting their investigation to the bivariate case d 1 = 1 = d 2 $$ {d}_1=1={d}_2 $$ . Their introduction briefly mentions that multivariate X $$ \mathbf{X} $$ and Y $$ \mathbf{Y} $$ simply should be replaced with their Euclidean norms ∥ X ∥ $$ \parallel \mathbf{X}\parallel $$ and ∥ Y ∥ $$ \parallel \mathbf{Y}\parallel $$ ; how to perform that replacement is not completely clear, though. Nor are the applicability and the performance of the [LRR] approach beyond the case of bivariate X , Y $$ \left(X,Y\right) $$ . Below, we discuss the possibility of extending distance covariance and [LRR]'s robustness results to X $$ \mathbf{X} $$ and Y $$ \mathbf{Y} $$ taking values in ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ , on (hyper)spheres, or on more general Riemannian manifolds. Actually, a rank-based extension of distance covariance for random vectors has been developed by Shi et al. (2022a, b) and Deb and Sen (2023), the results of whom we rapidly mention in Section 2. In Section 3, we further extend their multivariate approach to directional observations—taking values on (hyper)spheres. Extensions to more general compact Riemannian manifolds follow along similar lines, and we are currently working on it, but is not without some pitfalls. For instance, the equivalence between zero distance covariance and independence (the fundamental property justifying distance-covariance-based tests of independence in ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ ), in general, no longer holds for variables taking values on (hyper)spheres: see Section 3.2. The central idea of our approach is a combination of the distribution-free concepts of multivariate ranks and signs defined in Hallin et al. (2024) with distance covariance, via a suitable data transformation from hyperspheres to tangent spaces—a further illustration of the power of an adequate application of the ‘robustness by transformation’ idea developed in [LRR]. The distance covariance in Székely et al. (2007) was introduced for ℝ d 1 × ℝ d 2 $$ \left({\mathrm{\mathbb{R}}}^{d_1}\times {\mathrm{\mathbb{R}}}^{d_2}\right) $$ -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. A major difficulty stems from the fact that ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ , for d > 1 $$ d>1 $$ , is not canonically ordered, so that the very definition of ranks is problematic. None of the many attempts—marginal ranks, depth-based ranks, spatial ranks, Mahalanobis or elliptical ranks, … we refer to the online appendix of Hallin et al. (2021) for details and references—to provide a concept of rank in this multivariate context was fully satisfactory until measure-transportation-based definitions were considered. Chernozhukov et al. (2017) and, under the name of centre-outward distribution function, Hallin et al. (2021) consider the (a.s.) unique gradient of convex function F ± ≡ F Z ; ± $$ {\mathbf{F}}_{\pm}\equiv {\mathbf{F}}_{\mathbf{Z};\pm } $$ pushing the distribution P $$ \mathrm{P} $$ of some d $$ d $$ -dimensional random variable Z $$ \mathbf{Z} $$ forward to the spherical uniform U d $$ {\mathrm{U}}_d $$ over the unit ball S d $$ {\mathbbm{S}}_d $$ and show that it enjoys the typical properties of a distribution function. Based on a sample Z 1 n , … , Z n n $$ {\mathbf{Z}}_1^{(n)},\dots, {\mathbf{Z}}_n^{(n)} $$ of i.i.d. copies of Z ∼ P $$ \mathbf{Z}\sim \mathrm{P} $$ , Hallin et al. (2021) also propose empirical versions of F ± $$ {\mathbf{F}}_{\pm } $$ . For instance, the empirical optimal transport F 0 ; ± n $$ {\mathbf{F}}_{0;\pm}^{(n)} $$ of the sample to a grid of n $$ n $$ i.i.d. points with spherical uniform distribution U d $$ {\mathrm{U}}_d $$ is a consistent (in the Glivenko–Cantelli sense) estimator of F ± $$ {\mathbf{F}}_{\pm } $$ . But, for finite n $$ n $$ , F 0 ; ± n Z $$ {\mathbf{F}}_{0;\pm}^{(n)}\left(\mathbf{Z}\right) $$ does not enjoy the independence property of U ∼ U d $$ \mathbf{U}\sim {\mathrm{U}}_d $$ , namely, independence between ∥ U ∥ $$ \parallel \mathbf{U}\parallel $$ and U / ∥ U ∥ $$ \mathbf{U}/\parallel \mathbf{U}\parallel $$ : ∥ F 0 ; ± n Z ∥ $$ \parallel {\mathbf{F}}_{0;\pm}^{(n)}\left(\mathbf{Z}\right)\parallel $$ and F 0 ; ± n Z n / ∥ F 0 ; ± Z ∥ $$ {\mathbf{F}}_{0;\pm}^{(n)}{\left(\mathbf{Z}\right)}^{(n)}/\parallel {\mathbf{F}}_{0;\pm}\left(\mathbf{Z}\right)\parallel $$ are only asymptotically mutually independent. Also, F 0 ; ± n $$ {\mathbf{F}}_{0;\pm}^{(n)} $$ does not decompose into mutually independent ranks and signs and does not provide empirical quantile contours. A more structured grid, which involves a factorization of n $$ n $$ into n = n R n S + n 0 $$ n={n}_R{n}_S+{n}_0 $$ where n R $$ {n}_R $$ , n S $$ {n}_S $$ and n 0 $$ {n}_0 $$ are integers such that n 0 < min n R n S $$ {n}_0<\min \left({n}_R,{n}_S\right) $$ , provides an empirical and still Glivenko–Cantelli-consistent (as both n R $$ {n}_R $$ and n S $$ {n}_S $$ tend to infinity) version F ± n $$ {\mathbf{F}}_{\pm}^{(n)} $$ of F ± $$ {\mathbf{F}}_{\pm } $$ . Contrary to F 0 ; ± n $$ {\mathbf{F}}_{0;\pm}^{(n)} $$ , F ± n $$ {\mathbf{F}}_{\pm}^{(n)} $$ enjoys, for any n $$ n $$ (and with a tie-breaking procedure in case n 0 > 1 $$ {n}_0>1 $$ ) the independence property of U ∼ U d $$ \mathbf{U}\sim {\mathrm{U}}_d $$ . Hallin et al. (2021) then define the rank and the sign of Z i n $$ {\mathbf{Z}}_i^{(n)} $$ as R i n ≔ n R + 1 ∥ F ± n Z i n ∥ $$ {R}_i^{(n)}:= \left({n}_R+1\right)\parallel $$ and S i n ≔ F ± n Z i n / ∥ F ± n Z i n ∥ $$ $$ , We refer to Hallin et al. (2021) for details and to Hallin for a F 0 ; ± n $$ {\mathbf{F}}_{0;\pm}^{(n)} $$ for F ± n $$ {\mathbf{F}}_{\pm}^{(n)} $$ , alternative versions of d X n Y n $$ $$ and d J 1 , J 2 X n Y n $$ $$ are obtained in an not fully the of distance covariance statistics. the distance covariance between random vectors rank-based versions have been studied from the of of the bivariate results developed by [LRR] be a natural of their In redescending influence functions are to from the of their ( ψ + $$ $$ is the of in we suggest d J 1 , J 2 X n Y n $$ $$ distance covariance between a 2 d 1 $$ $$ -dimensional random and a 2 d 2 $$ $$ -dimensional as a of d J 1 , J 2 X n Y n $$ $$ is a distance covariance between a d 1 $$ $$ and a d 2 $$ $$ -dimensional random that the of the transformation is of by the fact that the J ∥ F ± ∥ $$ $$ , in are by the F ± / ∥ F ± ∥ $$ {\mathbf{F}}_{\pm $$ which are uniform over the unit This for the same of J 1 $$ {J}_1 $$ and J 2 $$ {J}_2 $$ as versions of the der Wilcoxon and rank-based distance We that the corresponding influence functions are and Section 2 of distance for multivariate real data. are and have not been so In this we are a extension to directional that is, to random on We that to other compact Riemannian such as and of are the same along with the for Riemannian manifolds developed by Hallin and we the to the case of with values on the in a of in of or of an on the of the and of for the by and The and difficulty with such data in the of their sample hyperspheres or are manifolds. this of data a of the problem of testing independence between two directional variables has been and some of the such as in and for their theoretical are The ‘robustness by data transformation in [LRR] the to and independence testing In this we discuss some we are the to such some by S d − 1 $$ $$ the unit in ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ . The distance between 1 $$ $$ and 2 $$ $$ in S d − 1 $$ $$ is d S 1 2 = 1 2 $$ $$ . with the S d − 1 $$ $$ is a a with $$ $$ d − 1 $$ $$ , The P Z $$ $$ is used for the probability distribution of the S d − 1 $$ $$ -valued random Z $$ \mathbf{Z} $$ . a Riemannian $$ $$ and a $$ $$ the S d − 1 $$ $$ ), by $$ $$ the tangent of $$ $$ at $$ $$ , and by $$ $$ the of $$ $$ . In the two we show how [LRR]'s robustness by transformation combined with the of optimal on hyperspheres on compact Riemannian and the introduction of directional ranks and provides a to these two with a definition of the concepts to be The of which can be to and has been applied in a of to mention only a for Hallin et al. et al. and In on Chernozhukov et al. (2017) the name of and Hallin et al. (2021) the name of centre-outward of ranks (and signs and in ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ that enjoy the properties from such which (for the ranks and the The approach and the we are is that of Hallin et al. measure-transportation-based have been used to distribution-free tests of independence in Euclidean in rank-based distance covariance test statistics for see Section Based on optimal in the context are Hallin et al. (2024) on hyperspheres and Hallin and (2024) on more general compact Riemannian manifolds also define distribution-free ranks and The directional distance covariance and the testing we are are on these theoretical and 1 in Hallin et al. (2024) the and P $$ \mathrm{P} $$ of the S P , $$ $$ , and 2 that it is a is between S d − 1 $$ $$ and S d − 1 $$ $$ ; call distance between P $$ \mathrm{P} $$ and $$ $$ the S P , $$ $$ of the $$ $$ is the uniform P U $$ $$ over S d − 1 $$ $$ , call F = F P $$ $$ and = P ≔ F P − 1 $$ $$ the directional distribution function and the directional quantile function of to F P $$ $$ and P $$ $$ , the F Z $$ $$ and Z $$ $$ be used in an for Z ∼ P $$ \mathbf{Z}\sim \mathrm{P} $$ ). definitions are a natural extension of the classical where the distribution function F $$ F $$ of a random variable Z ∼ P Z $$ $$ is pushing P Z $$ $$ forward to the uniform U 0 , 1 $$ $$ over the unit Z n = Z 1 n … Z n n $$ $$ a sample of n $$ n $$ i.i.d. copies of Z ∼ P $$ \mathbf{Z}\sim \mathrm{P} $$ . in ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ , two empirical versions of F ≡ F P $$ $$ , F 0 n $$ $$ and F n $$ $$ , can be with the that F n $$ $$ is the of a procedure of which F 0 n $$ $$ is the F 0 n $$ $$ and F n $$ $$ are Glivenko–Cantelli but only F n $$ $$ quantile ranks and the of F n $$ $$ . The involves the of a $$ $$ for P $$ \mathrm{P} $$ instance, the empirical n = n $$ $$ with an F 0 n $$ $$ of F $$ $$ . The is obtained as the mapping from the sample Z n $$ $$ to an adequate grid 0 n = 0 ; 1 n … 0 ; n n S d − 1 $$ $$ over the n $$ $$ of $$ $$ of the integers 1 , … , n $$ $$ , i = 1 n Z i n 0 ; i n $$ $$ $$ $$ defined in The F 0 n $$ $$ of this problem is the optimal transport pushing the empirical distribution of the sample forward to the empirical distribution of the grid 0 n $$ $$ . The only required for the of F 0 n $$ $$ is the as n $$ $$ , of the empirical distribution over 0 n $$ $$ to the uniform over S d − 1 $$ $$ ; to the that 0 n $$ $$ is obtained as an i.i.d. n $$ n $$ of uniform over S d − 1 $$ $$ This is an optimal problem for which are The resulting F 0 n $$ $$ does not for the of empirical quantile and ranks or it a n ≔ F 0 n n $$ $$ for the which the grid to be used in the be and provides Z i n $$ {\mathbf{Z}}_i^{(n)} $$ with a 1 − F 0 n ( Z i n ) , n $$ $$ and a of the n $$ $$ and F 0 n Z i n $$ $$ with the with n $$ $$ , that is, the d − 2 $$ $$ S d − 2 $$ $$ of the by n $$ $$ ; this with the n S d − 1 − n $$ $$ of the tangent at n $$ $$ . The involves a grid, n n $$ $$ , as in ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ , on a factorization of n $$ n $$ into n R n S + n 0 $$ $$ with n 0 < min n R n S $$ {n}_0<\min \left({n}_R,{n}_S\right) $$ . This grid of the n R n S $$ $$ of n S $$ {n}_S $$ n $$ $$ and − n $$ $$ ) with a uniform of n R $$ {n}_R $$ is the of points with along with n 0 $$ {n}_0 $$ copies of the n $$ $$ . by F n ≡ F Z n $$ $$ the optimal transport from the empirical distribution of the sample to the empirical distribution over this The directional ranks R i n $$ $$ values 1 , … , n R $$ {n}_R $$ ) and signs S i n $$ $$ values in the d − 2 $$ $$ of the by n $$ $$ ) to be used can be defined as functions of F Z n Z i n $$ $$ . We refer to Section of Hallin et al. (2024) for show how the ranks and signs obtained by Z i n = X i n $$ $$ and Z i n = Y i n $$ $$ in the in Section for distance covariance statistics that of the two and the fact that the is not of in Section 3.2. Now, the data and still values in S d 1 − 1 $$ $$ and S d 2 − 1 $$ $$ , which many of to be of the distance between them, this we propose a further transformation more a of further bijective transformations of F X n X i n $$ $$ and F Y n Y i n $$ $$ F 0 ; X n X i n $$ $$ and F 0 ; Y n Y i n $$ $$ ), i = 1 , … , n $$ n $$ mapping S d 1 − 1 $$ $$ and S d 2 − 1 $$ $$ to Euclidean these transformations are the null hypothesis of independence, for a of distance 1 $$ $$ and 2 $$ $$ the X i n , Y i n with values in S d 1 − 1 − 1 × S d 2 − 1 − 2 $$ $$ to d 1 − 1 × d 2 − 1 1 S d 1 − 1 × 2 S d 2 − 1 $$ $$ , which is a of Euclidean over which dCov $$ $$ is defined as for 1 $$ $$ and 2 $$ $$ are the empirical X n $$ $$ and Y n $$ $$ of 1 $$ $$ and 2 $$ $$ with any − X i n $$ $$ and − Y i n $$ $$ are to be the corresponding is We then propose the definition of a directional distance covariance distance the 1 2 $$ $$ distance covariance of X , Y $$ \left(\mathbf{X},\mathbf{Y}\right) $$ with values in S d 1 − 1 × S d 2 − 1 $$ $$ as bijective 1 $$ $$ and 2 $$ $$ from S d 1 − 1 − X n $$ $$ and S d 2 − 1 − Y n $$ $$ to d 1 − 1 $$ $$ of X n S d 1 − 1 $$ $$ and d 2 − 1 $$ $$ of Y n S d 2 − 1 $$ $$ , empirical versions of are the statistics d 1 , 2 X n Y n ≔ dCov 1 F X n X n , 2 F Y n Y n $$ $$ is and d 0 ; 1 , 2 X n Y n ≔ dCov 1 F 0 ; X n X n , 2 F 0 ; Y n Y n $$ $$ ; both are distribution-free under the null hypothesis ℋ 0 $$ {\mathrm{\mathscr{H}}}_0 $$ of independence between X $$ \mathbf{X} $$ and Y $$ \mathbf{Y} $$ . 1 , 2 n $$ $$ and 0 ; 1 , 2 n $$ $$ have their the one 0 ; 1 , 2 n $$ $$ is of 1 , 2 n $$ $$ it does not the optimal transport in 2 of Section the other 1 , 2 n $$ $$ , it does not fully as for the choice of 1 $$ $$ and 2 $$ $$ , one functions yielding robustness in the same as the function ψ $$ } $$ introduced in [LRR]. for R X , i n $$ $$ , S X , i n $$ $$ , R Y , i n $$ $$ , and S Y , i n $$ $$ ) where we that $$ $$ and $$ $$ are continuous and bijective functions mapping to ℝ $$ \mathrm{\mathbb{R}} $$ $$ $$ and $$ $$ are continuous and bijective from S d 1 − 2 $$ $$ and S d 2 − 2 $$ $$ to the tangent X n S d 1 − 1 $$ $$ and Y n S d 2 − 1 $$ $$ , for 1 , 2 n $$ $$ and 0 ; 1 , 2 n $$ $$ (as defined in by are distribution-free under the null hypothesis of independence. scores 1 u = 1 $$ $$ and 2 u = u $$ $$ for = 1 , 2 $$ $$ are they are an of a transformation of the yielding a test on the test 1 , 2 n $$ $$ ) that distribution-free but to some of Building on the measure-transportation-based of directional ranks and signs defined in Hallin et al. we show how the ‘robustness via transformation’ by [LRR] beyond the case of bivariate independence and also in Euclidean and on compact manifolds. The case of directional variables values on is are not of the we a of directional distance covariance test statistics and their We that this the to further in with [LRR]. by a as of the