SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE II. OPTIMAL R-ESTIMATION OF SHAPE

By Marc Hallin, Hannu Oja, Davy Paindaveine. 2006

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A class of R-estimators, based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in Hallin and Paindaveine (2006a), 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 thanthe 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 equivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasability and excellent finite-sample performances.

Subfields: Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/863e13b2061d7a2800862e7625e417a0842fb700

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Citations in the corpus

A canonical definition of shape (2008, unknown): “Classical choices include (i) S(Σ) = Σ11 (Hallin and Paindaveine (2006a), Hallin et al. (2006), Hettmansperger and Randles (2002), and Randles (2000)), (ii) S(Σ) = (trΣ)/k (Dümbgen (1998), Ollila et al. (2004), and Tyler (1987)), and (iii) S(Σ) = |Σ|1/k (Dümbgen and Tyler (2005), Hallin and…”
SEMIPARAMETRICALLY EFFICIENT RANK-BASED INFERENCE FOR SHAPE I. OPTIMAL RANK-BASED TESTS FOR SPHERICITY (2006, methodology): “1 of [16], or more general methods, such as the one recently developed by Andreou and Werker [2].”
Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach (2021, background): “…errors (Hallin and Paindaveine 2004a, 2005a, 2006a), VAR order identification (Hallin and Paindaveine 2004b), shape (Hallin and Paindaveine 2006b; Hallin, Oja, and Paindaveine 2006), homogeneity of scatter (Hallin and Paindaveine 2008), principal and common principal components (Hallin,…”
Elliptical graphical modelling (2011, background): “Assumption 3.1 is the important prerequisite on ˆ S n , and our results also apply to estimators that are asymptotically aﬃne equivariant, like the rank-based estimation technique of Hallin et al. (2006).”