Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent tests of independence between pairs of random variables. Using ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result, it has long remained unclear how one may construct distribution-free yet consistent tests of independence between random vectors. This is the problem addressed in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, and adopts a common standard form for dependence measures that encompasses many popular examples. In a unified study, we derive a general asymptotic representation of center-outward rank-based test statistics under independence, extending to the multivariate setting the classical Hájek asymptotic representation results. This representation permits direct calculation of limiting null distributions and facilitates a local power analysis that provides strong support for the center-outward approach by establishing, for the first time, the nontrivial power of center-outward rank-based tests over root-n neighborhoods within the class of quadratic mean differentiable alternatives.
The research reported in this manuscript is supported by the United States NSF Grants DMS-1712536 and SES-2019363 and European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 883818).
We acknowledge with thanks the stimulating comments by the Editor, Ming Yuan, the anonymous Associate Editor, and three anonymous referees, which led to a significant improvement of this manuscript.
Hongjian Shi. Marc Hallin. Mathias Drton. Fang Han. "On universally consistent and fully distribution-free rank tests of vector independence." Ann. Statist. 50 (4) 1933 - 1959, August 2022. https://doi.org/10.1214/21-AOS2151