We propose a family of tests, based on Randles' (1989) concept of interdirections and the ranks of pseudo-Mahalanobis distances computed with respect to a multivariate M-estimator of scatter due to Tyler (1987), for the multivariate one-sample problem under elliptical symmetry. These tests, which generalize the univariate signed-rank tests, are affine-invariant. Depending on the score function considered (van der Waerden, Laplace,...), they allow for locally asymptotically maximin tests at selected densities (multivariate normal, multivariate double-exponential,...). Local powers and asymptotic relative efficiencies are derived--with respect to Hotelling's test, Randles' (1989) multivariate sign test, Peters and Randles' (1990) Wilcoxon-type test, and with respect to the Oja median tests. We, moreover, extend to the multivariate setting two famous univariate results: the traditional Chernoff-Savage (1958) property, showing that Hotelling's traditional procedure is uniformly dominated, in the Pitman sense, by the van der Waerden version of our tests, and the celebrated Hodges-Lehmann (1956) ".864 result," providing, for any fixed space dimension $k$, the lower bound for the asymptotic relative efficiency of Wilcoxon-type tests with respect to Hotelling's.
These asymptotic results are confirmed by a Monte Carlo investigation, and application to a real data set.
"Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks." Ann. Statist. 30 (4) 1103 - 1133, August 2002. https://doi.org/10.1214/aos/1031689019