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December 2006 Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape
Marc Hallin, Hannu Oja, Davy Paindaveine
Ann. Statist. 34(6): 2757-2789 (December 2006). DOI: 10.1214/009053606000000948


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) 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 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 performance.


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Marc Hallin. Hannu Oja. Davy Paindaveine. "Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape." Ann. Statist. 34 (6) 2757 - 2789, December 2006.


Published: December 2006
First available in Project Euclid: 23 May 2007

zbMATH: 1115.62059
MathSciNet: MR2329466
Digital Object Identifier: 10.1214/009053606000000948

Primary: 62G35 , 62M15

Keywords: affine equivariance , Elliptical densities , local asymptotic normality , multivariate ranks and signs , one-step estimation , R-estimation , Semiparametric efficiency , shape matrix

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 6 • December 2006
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