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Semiparametrically efficient rankbased inference for shape I: Optimal rankbased tests for sphericity
 Ann. Statist
, 2006
"... A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under a ..."
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Cited by 48 (32 self)
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A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn 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 normaltheory 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 rankbased version of Le Cam’s onestep methodology which avoids the unpleasant nonparametric estimation of crossinformation quantities that is generally required in the context of Restimation. Although they are not strictly affineequivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finitesample performances. 1. Introduction. 1.1. Rankbased inference for elliptical families. An elliptical density over Rk is determined by a location center θ ∈ Rk, a scale parameter σ ∈ R + 0, a realvalued positive definite symmetric k × k matrix V = (Vij) with V11 = 1,
Equivariance and Invariance Properties of Multivariate Quantile and Related Functions, and the Role of Standardization
, 2009
"... Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invar ..."
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Cited by 17 (7 self)
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Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invariance of procedures, three important types of matrixvalued functional are studied: “weak covariance ” (or “shape”), “transformationretransformation ” (TR), and “strong invariant coordinate system ” (SICS). Clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artifacts of SICSstandardized data are invariant under affine transformation of the original data followed by standardization using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardization are described.
OPTIMAL RANKBASED TESTING FOR PRINCIPAL COMPONENTS
"... This paper provides parametric and rankbased optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudoGaussian robustifications by Tyler (1981, 1983) a ..."
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Cited by 13 (11 self)
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This paper provides parametric and rankbased optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudoGaussian robustifications by Tyler (1981, 1983) and Davis (1977), with which their Gaussian versions are shown to coincide, asymptotically, under Gaussian or finite fourthorder moment assumptions, respectively. Such assumptions however restrict the scope to covariancebased principal component analysis. The rankbased tests we are proposing remain valid without such assumptions. Hence, they address a much broader class of problems, where covariance matrices need not exist and principal components are associated with more general scatter matrices. Asymptotic relative efficiencies moreover show that those rankbased tests are quite powerful; when based on van der Waerden or normal scores, they even uniformly dominate the pseudoGaussian versions
OPTIMAL RANKBASED TESTS FOR HOMOGENEITY OF SCATTER
, 806
"... We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly r ..."
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Cited by 8 (8 self)
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We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavytailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normalscore version outperforms traditional Gaussian likelihood ratio tests and their pseudoGaussian robustifications under a very broad range of nonGaussian densities including, for instance, all multivariate Student and powerexponential distributions. 1. Introduction. 1.1. Homogeneity of variances and covariance matrices. The assumption of variance homogeneity is central to the theory and practice of univariate
Optimal tests for homogeneity of covariance, scale, and shape
 J. Multivariate Anal
, 2008
"... The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic cont ..."
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Cited by 7 (4 self)
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The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic context of multivariate Gaussian models. And the modified likelihood ratio tests (MLRT) that are used in everyday practice are known to be highly sensitive to violations of Gaussian assumptions. In this paper, we provide a complete and systematic study of the problem, and propose test statistics which, while preserving the optimality features of the MLRT under multinormal assumptions, remain valid under unspecified elliptical densities with finite fourthorder moments. As a first step, the Le Cam LAN approach is used for deriving locally and asymptotically optimal testing procedures φ (n) f for any specified mtuple of radial densities f = (f1,..., fm). Combined with an estimation of the m densities f1,..., fm, these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudoGaussian tests—namely, tests that are locally and asymptotically optimal at Gaussian densities while remaining valid under a much broader class of distributions—in general are preferable. We therefore construct two pseudoGaussian modifications of the Gaussian version φ (n) N of the optimal test φ (n) f. The first one, φ
On Invariant Coordinate System (ICS) Functionals
"... Summary. Equivariance and invariance issues often arise in multivariate statistical analysis. Statistical procedures have to be modified sometimes to obtain an affine equivariant or invariant version. This is often done by preprocessing the data, e.g., by standardizing the multivariate data or by tr ..."
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Cited by 4 (3 self)
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Summary. Equivariance and invariance issues often arise in multivariate statistical analysis. Statistical procedures have to be modified sometimes to obtain an affine equivariant or invariant version. This is often done by preprocessing the data, e.g., by standardizing the multivariate data or by transforming the data to an invariant coordinate system. In this article standardization of multivariate distributions, and characteristics of invariant coordinate system (ICS) functionals and statistics, are examined. Also, invariances up to some groups of transformations are discussed. Constructions of ICS functionals are addressed. In particular the construction based on the use of two scatter matrix functionals presented by Tyler et al. (2009), and direct definitions based on the approach presented by Chaudhuri and Sengupta (1993) and related approaches, are examined. Several applications of ICS functionals are also discussed.
and Common Principal Components
"... We propose rankbased estimators of principal components, both in the onesample and, under the assumption of common principal components, in the msample cases. Those estimators are obtained via a rankbased version of Le Cam’s onestep method, combined with an estimation of crossinformation quanti ..."
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We propose rankbased estimators of principal components, both in the onesample and, under the assumption of common principal components, in the msample cases. Those estimators are obtained via a rankbased version of Le Cam’s onestep method, combined with an estimation of crossinformation quantities. Under arbitrary elliptical distributions with, in the msample case, possibly heterogeneous radial densities, those Restimators remain rootn consistent and asymptotically normal, while achieving asymptotic efficiency under correctly specified densities. Contrary to their traditional counterparts computed from empirical covariances, they do not require any moment conditions. When based on Gaussian score functions, in the onesample case, they moreover uniformly dominate their classical
The Annals of Statistics OPTIMAL RANKBASED TESTS FOR HOMOGENEITY OF SCATTER
"... We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs, for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectl ..."
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We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs, for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavytailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normalscore version outperforms traditional Gaussian likelihood ratio tests and their pseudoGaussian robustifications under a very broad range of nonGaussian densities including, for instance, all multivariate Student and powerexponential distributions. ∗The authors are also members of ECORE, the recently created association between CORE and ECARES.
SPATIAL SIGN CORRELATION
, 2014
"... Abstract. A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimato ..."
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Abstract. A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimators by means of numerical simulations. 1.
by
, 2009
"... A generalization of Tyler's Mestimators to the case of incomplete data Discussion papers in statistics and econometrics, No. 3/07 Provided in Cooperation with: ..."
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A generalization of Tyler's Mestimators to the case of incomplete data Discussion papers in statistics and econometrics, No. 3/07 Provided in Cooperation with: