Semiparametrically efficient rank-based inference for shape II.: Optimal R-estimation of shape

• 成果类型: Article
• 署名作者: Hallin, Marc; Oja, Hannu; Paindaveine, Davy
• 署名单位: Universite Libre de Bruxelles; Universite Libre de Bruxelles; Tampere University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000948
• 发表日期: 2006
• 页码: 2757-2789
• 关键词: pseudo-mahalanobis ranks; regression coefficients; multivariate location; Asymptotic Normality; affine-invariant; optimal tests; sphericity; parameter; scatter; interdirections

摘要

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.