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

TL;DR

This paper introduces semiparametrically efficient rank-based estimators for the shape matrix of elliptical distributions, achieving high efficiency without moment assumptions and outperforming traditional covariance-based methods.

Contribution

It proposes a novel class of R-estimators based on multivariate signed ranks that are root-n consistent and semiparametrically efficient, avoiding complex nonparametric estimation.

Findings

Estimators are root-n consistent under any radial density.

They are semiparametrically efficient at a prespecified density.

Simulations show excellent finite-sample performance.

Abstract

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.