Chernoff-Savage and Hodges-Lehmann results for Wilks' test of multivariate independence

TL;DR

This paper extends classical univariate results to rank-based multivariate independence tests, showing the superiority of a normal-score rank test over Wilks' test and providing efficiency bounds.

Contribution

It introduces a rank-based test for multivariate independence that dominates Wilks' test and establishes efficiency bounds for the Wilcoxon version.

Findings

Normal-score rank test uniformly dominates Wilks' test

Wilcoxon version has quantifiable asymptotic efficiency bounds

Results establish Pitman non-admissibility of Wilks' test

Abstract

We extend to rank-based tests of multivariate independence the Chernoff-Savage and Hodges-Lehmann classical univariate results. More precisely, we show that the Taskinen, Kankainen and Oja (2004) normal-score rank test for multivariate independence uniformly dominates -- in the Pitman sense -- the classical Wilks (1935) test, which establishes the Pitman non-admissibility of the latter, and provide, for any fixed space dimensions $p,q$ of the marginals, the lower bound for the asymptotic relative efficiency, still with respect to Wilks' test, of the Wilcoxon version of the same.