Distribution-Free Consistent Independence Tests via Center-Outward Ranks and Signs

By Hongjian Shi, Mathias Drton, Fang Han. Journal of the American Statistical Association, 2019

89 citations recorded by the source corpus.

Abstract This article investigates the problem of testing independence of two random vectors of general dimensions. For this, we give for the first time a distribution-free consistent test. Our approach combines distance covariance with the center-outward ranks and signs developed by Marc Hallin and collaborators. In technical terms, the proposed test is consistent and distribution-free in the family of multivariate distributions with nonvanishing (Lebesgue) probability densities. Exploiting the (degenerate) U-statistic structure of the distance covariance and the combinatorial nature of Hallin’s center-outward ranks and signs, we are able to derive the limiting null distribution of our test statistic. The resulting asymptotic approximation is accurate already for moderate sample sizes and makes the test implementable without requiring permutation. The limiting distribution is derived via a more general result that gives a new type of combinatorial noncentral limit theorem for double- and multiple-indexed permutation statistics. Supplementary materials for this article are available online.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2bd60ac8124f60e56497ad0f60d791e836f2d392

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Citations in the corpus

Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, methodology): “In Section 8 we use the recently proposed idea of multivariate ranks based on the theory of optimal transport (see [50, 51, 25, 30, 89]) to develop measures of association that satisfy properties (I)-(III) and are also finite-sample distribution-free when μ = μX ⊗ μY ; see Sections 8.”
On Azadkia–Chatterjee’s conditional dependence coefficient (2021, background): “Here, λk, k ∈ Z+, are the non-zero eigenvalues of the integral equation given by Shi et al. (2022a, Equation (12)) and depend only on p+q, and [ξk] ∞ k=1 is a sequence of independent standard Gaussian random variables. Further details can be found in Shi et al. (2022a). By Theorem 3.1 in Shi et al. ”
Multivariate Rank-Based Distribution-Free Nonparametric Testing Using Measure Transportation (2019, methodology): “However, the asymptotic consistency results in Shi, Drton, and Han (2020) are derived under more stringent conditions (e.g., nonvanishing Lebesgue probability densities).”
On the power of Chatterjee’s rank correlation (2020, background): “This makes any subsequent local power analysis extremely difficult, as also observed in a parallel study of center-outward rank and sign–based tests performed in Shi et al. (2020a) and Shi et al. (2020b) (see, also, Deb and Sen (2019)).”