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34
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,
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
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
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.
Nonparametrically Consistent Depthbased Classifiers
"... We introduce a class of depthbased classification procedures that are of a nearestneighbor nature. Depth, after symmetrization, indeed provides the centeroutward ordering that is necessary and sufficient to define nearest neighbors. The resulting classifiers are affineinvariant and inherit the ..."
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Cited by 7 (0 self)
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We introduce a class of depthbased classification procedures that are of a nearestneighbor nature. Depth, after symmetrization, indeed provides the centeroutward ordering that is necessary and sufficient to define nearest neighbors. The resulting classifiers are affineinvariant and inherit the nonparametric validity from nearestneighbor classifiers. In particular, we prove that the proposed depthbased classifiers are consistent under very mild conditions. We investigate their finitesample performances through simulations and show that they outperform affineinvariant nearestneighbor classifiers obtained through an obvious standardization construction. We illustrate the practical value of our classifiers on two real data examples. Finally, we shortly discuss the possible uses of our depthbased neighbors in other inference problems.
Robustness of weighted Lpdepth and Lpmedian
 Allgemeines Statistisches Archiv
, 2004
"... Summary: Lpnorm weighted depth functions are introduced and the local and global robustness of these weighted Lpdepth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a not ..."
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Cited by 4 (1 self)
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Summary: Lpnorm weighted depth functions are introduced and the local and global robustness of these weighted Lpdepth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted Lpdepth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted Lpdepth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted Lpmedians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted Lp depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted Lp depth and medians, respectively.
An Affine Invariant kNearest Neighbor Regression Estimate
, 1201
"... We design a datadependent metric in R d and use it to define the knearest neighbors of a given point. Our metric is invariant under all affine transformations. We show that, with this metric, the standard knearestneighborregression estimateisasymptotically consistent under the usual conditions on ..."
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Cited by 4 (0 self)
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We design a datadependent metric in R d and use it to define the knearest neighbors of a given point. Our metric is invariant under all affine transformations. We show that, with this metric, the standard knearestneighborregression estimateisasymptotically consistent under the usual conditions on k, and minimal requirements on the input data. Index Terms — Nonparametric estimation, Regression function estimation,
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.