Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity

A class of R-estimators based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in Hallin and Painda-veine [Ann. Statist. 34 (2006) 2707–2756] 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 nor-mality of which, moreover, requires finite moments of order four), irrespec-tive of the actual underlying elliptical density. They rely on an original rank-based version of Le Cam’s one-step methodology which avoids the unpleas-ant 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 performance