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Semiparametrically efficient rank-based inference for shape. II: Optimal \(R\)-estimation of shape. (English) Zbl 1115.62059

Summary: A class of \(R\)-estimators based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in part I of this paper by M. Hallin and D. Paindaveine [ibid., 2707–2756 (2006; Zbl 1114.62066)] 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 L. Le Cam’s one-step methodology [see “Asymptotic methods in statistical decision theory.” (1986; Zbl 0605.62002)] which avoids the unpleasant nonparametnc 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.

MSC:

62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
62G10 Nonparametric hypothesis testing
60F05 Central limit and other weak theorems
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