Due to the lack of a canonical ordering in ${\mathbb R}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to a grid that approximates a uniform reference measure (Chernozhukov et al., 2017; Hallin, 2017; Hallin et al., 2021), thereby inducing ranks, signs, and a data-driven ordering of ${\mathbb R}^d$. We take up this new perspective to define and study multivariate analogues of the sign covariance/quadrant statistic, Spearman's rho, Kendall's tau, and van der Waerden covariances. The resulting tests of multivariate independence are fully distribution-free, hence uniformly valid irrespective of the actual (absolutely continuous) distribution of the observations. Our results provide the asymptotic distribution theory for these new test statistics, with asymptotic approximations to critical values to be used for testing independence between random vectors, as well as a power analysis of the resulting tests in an extension of the so-called Konijn model. For the van der Waerden tests, this power analysis includes a multivariate Chernoff--Savage property guaranteeing that, under elliptical generalized Konijn models, the asymptotic relative efficiency with respect to Wilks' classical (pseudo-)Gaussian procedure of our van der Waerden tests is strictly larger than or equal to one, where equality is achieved under Gaussian distributions only. We similarly provide a lower bound for the asymptotic relative efficiency of our Spearman procedure with respect to Wilks' test, thus extending the classical result by Hodges and Lehmann on the asymptotic relative efficiency, in univariate location models, of Wilcoxon tests with respect to the Student ones.

翻译：由于在$d>1$的${\mathbb R}^d$中缺乏规范排序，定义经典单变量秩的多元推广一直是统计学中一个长期悬而未决的问题。最优传输已被证明提供了一种解决方案，通过将数据点传输到近似均匀参考度量的网格上来获得多元秩（Chernozhukov等，2017；Hallin，2017；Hallin等，2021），从而生成${\mathbb R}^d$的秩、符号和数据驱动的排序。我们采用这一新视角来定义并研究符号协方差/象限统计量、斯皮尔曼rho、肯德尔tau和范德瓦登协方差的多元类似量。由此产生的多元独立性检验完全是无分布的，因此无论观测值的实际（绝对连续）分布如何，都具有一致有效性。我们的结果为这些新检验统计量提供了渐近分布理论，给出了用于检验随机向量之间独立性的临界值的渐近近似，并在所谓Konijn模型的扩展中分析了这些检验的势。对于范德瓦登检验，该势分析包括多元Chernoff-Savage性质，保证在椭圆广义Konijn模型下，我们的范德瓦登检验相对于Wilks经典（伪）高斯方法的渐近相对效率严格大于或等于1，且仅在高斯分布下达到相等。我们同样为斯皮尔曼检验相对于Wilks检验的渐近相对效率提供了下界，从而将Hodges和Lehmann关于单变量位置模型中Wilcoxon检验相对于Student检验渐近相对效率的经典结果进行了推广。