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12
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,
A canonical definition of shape
, 2007
"... Very general concepts of scatter, extending the traditional notion of covariance matrices, have become classical tools in robust multivariate analysis. In many problems of practical importance (principal components, canonical correlation, testing for sphericity), only homogeneous functions of the ..."
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Cited by 11 (6 self)
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Very general concepts of scatter, extending the traditional notion of covariance matrices, have become classical tools in robust multivariate analysis. In many problems of practical importance (principal components, canonical correlation, testing for sphericity), only homogeneous functions of the scatter matrix are of interest. In line with this fact, scatter functionals often are only defined up to a positive scalar factor, yielding a family of scatter matrices rather than a uniquely defined one. In such families, it is natural to single out one representative by imposing a normalization constraint: this normalized scatter is called a shape matrix. In the particular case of elliptical families, this constraint in turn induces a concept of scale; along with a location center and a standardized radial density, the shape and scale parameters entirely characterize an elliptical density. In this paper, we show that one and only normalization has the additional properties that (i) the resulting Fisher information matrices for shape and scale, in locally asymptotically normal (LAN) elliptical families, are blockdiagonal, and that (ii) the semiparametric elliptical families indexed by location, shape, and completely unspecified radial densities are adaptive. This particular normalization, which imposes that the determinant of the shape matrix be equal to one, therefore can be considered canonical.
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
The influence of observations on misclassification probability in multiple discriminant analysis
 Communications in Statistics. Theory and Methods
, 1996
"... In this paper it is studied how observations in the training sample affect the misclassification probability of a quadratic discriminant rule. An approach based on partial influence functions is followed. It allows to quantify the effect of observations in the training sample on the performance of ..."
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Cited by 7 (3 self)
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In this paper it is studied how observations in the training sample affect the misclassification probability of a quadratic discriminant rule. An approach based on partial influence functions is followed. It allows to quantify the effect of observations in the training sample on the performance of the associated classification rule. Focus is on the effect of outliers on the misclassification rate, merely than on the estimates of the parameters of the quadratic discriminant rule. The expression for the partial influence function is then used to construct a diagnostic tool for detecting influential observations. Applications on real data sets are provided.
Sign and rank covariance matrices: statistical properties and application to principal components analysis
 Statistical Data Analysis Based on the L1norm and Related
, 2002
"... Abstract. In this paper, the estimation of covariance matrices based on multivariate sign and rank vectors is discussed. Equivariance and robustness properties of the sign and rank covariance matrices are described. We show their use for the principal components analysis (PCA) problem. Limiting ef ..."
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Cited by 7 (1 self)
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Abstract. In this paper, the estimation of covariance matrices based on multivariate sign and rank vectors is discussed. Equivariance and robustness properties of the sign and rank covariance matrices are described. We show their use for the principal components analysis (PCA) problem. Limiting efficiencies of the estimation procedures for PCA are compared. 1.
Asymptotic linearity of serial and nonserial multivariate signed rank statistics
 Journal of Statistical Planning and Inference
, 2005
"... Asymptotic linearity plays a key role in estimation and testing in the presence of nuisance parameters. This property is established, in the very general context of a multivariate general linear model with elliptical VARMA errors, for the serial and nonserial multivariate rank statistics considered ..."
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Cited by 6 (4 self)
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Asymptotic linearity plays a key role in estimation and testing in the presence of nuisance parameters. This property is established, in the very general context of a multivariate general linear model with elliptical VARMA errors, for the serial and nonserial multivariate rank statistics considered in Hallin and Paindaveine (2002a and b, 2004a) and Oja and Paindaveine (2004). These statistics, which are multivariate versions of classical signed rank statistics, involve (i) multivariate signs based either on (pseudo)Mahalanobis residuals, or on a modified version (absolute interdirections) of Randles’s interdirections, and (ii) a concept of ranks based either on (pseudo)Mahalanobis distances or on liftinterdirections.
The Annals of Statistics SEMIPARAMETRICALLY EFFICIENT RANKBASED INFERENCE FOR SHAPE I. OPTIMAL RANKBASED TESTS FOR SPHERICITY
"... We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. T ..."
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We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations, and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distributionfree when the center of symmetry is specified, and asymptotically so, when it has to be estimated. The multivariate ranks used throughout are those of the distances—in the metric associated with the null value V0 of the shape matrix—between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and
The Annals of Statistics OPTIMAL RANKBASED TESTS FOR SPHERICITY
"... We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. T ..."
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We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations, and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distributionfree when the center of symmetry is specified, and asymptotically so, when it has to be estimated. The multivariate ranks used throughout are those of the distances—in the metric associated with the null value V0 of the shape matrix—between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and
www.elsevier.com/locate/jmva Symmetrised Mestimators of multivariate scatter
, 2005
"... In this paper we introduce a family of symmetrised Mestimators of multivariate scatter. These are defined to be Mestimators only computed on pairwise differences of the observed multivariate data. Symmetrised Huber’s Mestimator and Dümbgen’s estimator serve as our examples. The influence function ..."
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In this paper we introduce a family of symmetrised Mestimators of multivariate scatter. These are defined to be Mestimators only computed on pairwise differences of the observed multivariate data. Symmetrised Huber’s Mestimator and Dümbgen’s estimator serve as our examples. The influence functions of the symmetrised Mfunctionals are derived and the limiting distributions of the estimators are discussed in the multivariate elliptical case to consider the robustness and efficiency properties of estimators. The symmetrised Mestimators have the important independence property; they can therefore be used to find the independent components in the independent component analysis (ICA).