On Universally Consistent and Fully Distribution-Free Rank Tests of Vector Independence

• Activity Number: 530 - Recent Advances in Rank-Based Inference
• Type: Invited
• Date/Time: Thursday, August 11, 2022 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Hongjian Shi* and Marc Hallin and Mathias Drton and Fang Han
• Companies: Technical University of Munich and Université libre de Bruxelles and Technical University of Munich and University of Washington
• Keywords: multivariate ranks and signs; center-outward ranks and signs; multivariate dependence measure; independence test; Hájek representation; Le Cam’s third lemma

Abstract

Rank correlations have found many innovative applications in the last decade. In particular, suitable rank correlations have been used for consistent and distribution-free tests of independence between pairs of random variables. 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. In this talk we will discuss how this problem can be addressed via a general framework for designing multivariate dependence measures and associated test statistics based on the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks. In this framework we obtain new multivariate Hájek asymptotic representation results and use them for local power analyses that demonstrate the statistical efficiency of our tests.