[Paper Review] Distribution-free consistent independence tests via center-outward ranks and signs

Hongjian Shi, Mathias Drton+1 more|arXiv (Cornell University)|Sep 22, 2019

Bayesian Methods and Mixture Models32 references4 citations

TL;DR

This paper introduces the first distribution-free consistent test for independence between two multivariate random vectors of arbitrary dimensions, combining distance covariance with center-outward ranks and signs. The method achieves asymptotically accurate null distribution approximation without permutation, enabling direct implementation and demonstrating strong power even in high dimensions with moderate sample sizes.

ABSTRACT

This paper 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 in Hallin (2017). 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 non-central limit theorem for double- and multiple-indexed permutation statistics.

Motivation & Objective

• To develop a consistent independence test for multivariate random vectors that is distribution-free under general conditions.
• To overcome the limitation of existing distance covariance and kernel-based tests, which require permutation due to distribution-dependent null distributions.
• To integrate center-outward ranks and signs—previously used in univariate settings—into a multivariate independence framework.
• To derive a limiting null distribution for the test statistic that is both analytically tractable and accurate for moderate sample sizes.
• To establish a new combinatorial non-central limit theorem for double- and multiple-indexed permutation statistics to support theoretical results.

Proposed method

• The test combines distance covariance with center-outward ranks and signs, which generalize univariate ranks to multivariate settings via spherical symmetry and data depth.
• It leverages the degenerate U-statistic structure of distance covariance and the combinatorial nature of center-outward ranks to derive the limiting null distribution.
• The limiting distribution is derived via a novel combinatorial non-central limit theorem for permutation statistics with double and multiple indices.
• The test statistic is constructed from centered, rank-based distances, ensuring invariance under monotonic transformations and distribution-freeness under the null.
• Critical values are numerically estimated for various dimensions (p,q) up to (10,10) at significance levels α=0.1, 0.05, and 0.01, enabling direct implementation.
• The method avoids permutation by relying on the derived asymptotic null distribution, which is shown to be accurate even for moderate sample sizes.

Experimental results

Research questions

• RQ1 Can a distribution-free, consistent test for multivariate independence be constructed that does not require permutation?
• RQ2 How can center-outward ranks and signs be extended to multivariate settings to enable distribution-free testing?
• RQ3 What is the limiting null distribution of a distance covariance-based test using center-outward ranks in high-dimensional settings?
• RQ4 Can a new combinatorial non-central limit theorem be established to support the asymptotic theory of such permutation statistics?
• RQ5 How does the proposed test compare in power to existing methods like distance covariance with marginal ranks or ranked distance covariance?

Key findings

• The proposed test is the first to be both distribution-free and consistent for testing independence between two multivariate random vectors of general dimensions.
• The limiting null distribution of the test statistic is derived analytically and shown to be accurate for moderate sample sizes, eliminating the need for permutation.
• Numerical critical values are provided for all (p,q) pairs from (1,1) to (10,10) at α=0.1, 0.05, and 0.01, enabling direct implementation.
• In simulation studies, the test maintains high power even at high dimensions (p=q=7) with n=216, outperforming competitors in detecting complex dependence structures.
• The test via distance covariance with marginal ranks achieved the highest power in Example C.2, but the proposed method remained competitive and powerful when n≥216.
• The method demonstrates sensitivity to dimensionality, with power increasing significantly as dimension grows, especially when sample size is sufficient.