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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
Central limit theorem and influence function for the MCD estimators at general multivariate distributions. Bernouilli , forthcoming
, 2011
"... We define the minimum covariance determinant functionals for multivariate location and scatter through trimming functions and establish their existence at any multivariate distribution. We provide a precise characterization including a separating ellipsoid property and prove that the functionals ar ..."
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Cited by 8 (1 self)
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We define the minimum covariance determinant functionals for multivariate location and scatter through trimming functions and establish their existence at any multivariate distribution. We provide a precise characterization including a separating ellipsoid property and prove that the functionals are continuous. Moreover we establish asymptotic normality for both the location and covariance estimator and derive the influence function. These results are obtained in a very general multivariate setting. 1
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
Asymptotic expansion of the minimum covariance determinant estimators
, 2009
"... In Cator and Lopuhaä [3] an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and nonsingularity of the derivative in a firstorder Taylor expansion. In this paper, we prove the existence of this derivative for multivariate ..."
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Cited by 7 (2 self)
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In Cator and Lopuhaä [3] an asymptotic expansion for the MCD estimators is established in a very general framework. This expansion requires the existence and nonsingularity of the derivative in a firstorder Taylor expansion. In this paper, we prove the existence of this derivative for multivariate distributions that have a density and provide an explicit expression. Moreover, under suitable symmetry conditions on the density, we show that this derivative is nonsingular. These symmetry conditions include the elliptically contoured multivariate locationscatter model, in which case we show that the minimum covariance determinant (MCD) estimators of multivariate location and covariance are asymptotically equivalent to a sum of independent identically distributed vector and matrix valued random elements, respectively. This provides a proof of asymptotic normality and a precise description of the limiting covariance structure for the MCD estimators. 1
Robust Location and Scatter Estimators in Multivariate Analysis
"... The sample mean vector and the sample covariance matrix are the corner stone of the classical multivariate analysis. They are optimal when the underlying data are normal. They, however, are notorious for being extremely sensitive to outliers and heavy tailed noise data. This article surveys robust a ..."
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Cited by 3 (0 self)
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The sample mean vector and the sample covariance matrix are the corner stone of the classical multivariate analysis. They are optimal when the underlying data are normal. They, however, are notorious for being extremely sensitive to outliers and heavy tailed noise data. This article surveys robust alternatives of these classical location and scatter estimators and discusses their applications to the multivariate data analysis. 1.
unknown title
, 2009
"... Central limit theorem and influence function for the MCD estimators at general multivariate distributions ..."
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Central limit theorem and influence function for the MCD estimators at general multivariate distributions
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.
Printed in Great Britain A weighted multivariate sign test for clustercorrelated data
"... We consider the multivariate location problem with clustercorrelated data. A family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights. Under weak assumptions, the test statistic is asymptotically distributed as a chisquar ..."
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We consider the multivariate location problem with clustercorrelated data. A family of multivariate weighted sign tests is introduced for which observations from different clusters can receive different weights. Under weak assumptions, the test statistic is asymptotically distributed as a chisquared random variable as the number of clusters goes to infinity. The asymptotic distribution of the test statistic is also given for a local alternative model under multivariate normality. Optimal weights maximizing Pitman asymptotic efficiency are provided. These weights depend on the cluster sizes and on the intracluster correlation. Several approaches for estimating these weights are presented. Using Pitman asymptotic efficiency, we show that appropriate weighting can increase substantially the efficiency compared to a test that gives the same weight to each cluster. A multivariate weighted ttest is also introduced. The finitesample performance of the weighted sign test is explored through a simulation study which shows that the proposed approach is very competitive. A real data example illustrates the practical application of the methodology.