  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.
