  A class of R-estimators based on the concepts of multivariate signed ranks
and the optimal rank-based tests developed in Hallin and Paindaveine [Ann.
Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an
elliptical distribution. These R-estimators are root-n consistent under any
radial density g, without any moment assumptions, and semiparametrically
efficient at some prespecified density f. When based on normal scores, they are
uniformly more efficient than the traditional normal-theory estimator based on
empirical covariance matrices (the asymptotic normality of which, moreover,
requires finite moments of order four), irrespective of the actual underlying
elliptical density. They rely on an original rank-based version of Le Cam's
one-step methodology which avoids the unpleasant nonparametric estimation of
cross-information quantities that is generally required in the context of
R-estimation. Although they are not strictly affine-equivariant, they are shown
to be equivariant in a weak asymptotic sense. Simulations confirm their
feasibility and excellent finite-sample performances.
