Symmetric rank covariances: a generalized framework for nonparametric measures of dependence

By Luca Weihs, Mathias Drton, Nicolai Meinshausen. 2017

76 citations recorded by the source corpus.

The need to test whether two random vectors are independent has spawned a large number of competing measures of dependence. We are interested in nonparametric measures that are invariant under strictly increasing transformations, such as Kendall's tau, Hoeffding's D, and the more recently discovered Bergsma--Dassios sign covariance. Each of these measures exhibits symmetries that are not readily apparent from their definitions. Making these symmetries explicit, we define a new class of multivariate nonparametric measures of dependence that we refer to as Symmetric Rank Covariances. This new class generalises all of the above measures and leads naturally to multivariate extensions of the Bergsma--Dassios sign covariance. Symmetric Rank Covariances may be estimated unbiasedly using U-statistics for which we prove results on computational efficiency and large-sample behavior. The algorithms we develop for their computation include, to the best of our knowledge, the first efficient algorithms for the well-known Hoeffding's D statistic in the multivariate setting.

Subfields: Statistics and Probability

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Citations in the corpus

On the power of Chatterjee’s rank correlation (2020, methodology): “Proposition 2.2(ii)–(iv) still hold for all bivariate distributions (Proposition 1 in Weihs et al., 2018b and Theorem 5.4.”
High-dimensional consistent independence testing with maxima of rank correlations (2018, background): “= 3/(π 4i2j2), i, j ∈ Z+; see, e.g., Proposition 7 in Weihs et al. (2018) or Theorem 4.4 in Nandy et al. (2016).”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, background): “In fact, RdCov 2 n is exactly equivalent to the statistic proposed in [151] (also see the right sides of (4.”
The conditional permutation test for independence while controlling for confounders (2018, methodology): “based on ranks, kernel methods, copulas and nearest neighbours (Weihs et al., 2018; Pfister et al., 2018; Kojadinovic and Holmes, 2013; Berrett and Samworth, 2019).”
Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs (2019, background): “Weihs, Drton, and Meinshausen (2018) proposed other multivariate extensions of Hoeffding’s D, Blum–Kiefer–Rosenblatt’s R, and Bergsma–Dassios– Yanagimoto’s τ ∗, and did numerical studies comparing them to distance covariance applied to marginal ranks.”
The conditional permutation test (2018, background): “Simulta-neously, a great deal of research eﬀort has gone into developing several diﬀerent tests of independence, for example based on ranks, kernel methods, copulas, and nearest neighbours [26, 13, 12, 4].”
A New Coefficient of Correlation (2019, methodology): “There have been many proposals to address this deficiency of the classical coefficients [23], such as the maximal correlation coefficient [5, 15, 21, 33], various coefficients based on joint cumulative distribution functions and ranks [3, 4, 8, 9, 11, 14, 18, 22, 30, 32, 35, 36, 46–49], kernel-based”
On boosting the power of Chatterjee’s rank correlation (2021, background): “…al., 1961; Yanagimoto, 1970; Bergsma and Dassios, 2014; Shi et al., 2021b), the correlation measure ξ is to us appealing due to its consistency against dependence (Weihs et al., 2018) as well as capability of detecting perfect functional dependence (Cao and Bickel, 2020), which we summarize below.”