  The Wilcoxon rank-sum test is one of the most popular distribution-free
procedures for testing the equality of two univariate probability
distributions. One of the main reasons for its popularity can be attributed to
the remarkable result of Hodges and Lehmann (1956), which shows that the
asymptotic relative efficiency of Wilcoxon's test with respect to Student's
$t$-test, under location alternatives, never falls below 0.864, despite the
former being exactly distribution-free for all sample sizes. Even more striking
is the result of Chernoff and Savage (1958), which shows that the efficiency of
a Gaussian score transformed Wilcoxon's test, against the $t$-test, is lower
bounded by 1. In this paper we study the two-sample problem in the multivariate
setting and propose distribution-free analogues of the Hotelling $T^2$ test
(the natural multidimensional counterpart of Student's $t$-test) based on
optimal transport and obtain extensions of the above celebrated results over
various natural families of multivariate distributions. Our proposed tests are
consistent against a general class of alternatives and satisfy Hodges-Lehmann
and Chernoff-Savage-type efficiency lower bounds, despite being entirely
agnostic to the underlying data generating mechanism. In particular, a
collection of our proposed tests suffer from no loss in asymptotic efficiency,
when compared to Hotelling $T^2$. To the best of our knowledge, these are the
first collection of multivariate, nonparametric, exactly distribution-free
tests that provably achieve such attractive efficiency lower bounds. We also
demonstrate the broader scope of our methods in optimal transport based
nonparametric inference by constructing exactly distribution-free multivariate
tests for mutual independence, which suffer from no loss in asymptotic
efficiency against the classical Wilks' likelihood ratio test, under Konijn
alternatives.
