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48
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 12 (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 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 11 (9 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 Chernoff–Savage result for shape. On the nonadmissibility of pseudoGaussian methods
 J. Multivariate Anal
, 2006
"... Chernoff and Savage (1958) established that, in the context of univariate location models, Gaussianscore rankbased procedures uniformly dominate—in terms of Pitman asymptotic relative efficiencies—their pseudoGaussian parametric counterparts. This result, which had quite an impact on the success ..."
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Cited by 10 (8 self)
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Chernoff and Savage (1958) established that, in the context of univariate location models, Gaussianscore rankbased procedures uniformly dominate—in terms of Pitman asymptotic relative efficiencies—their pseudoGaussian parametric counterparts. This result, which had quite an impact on the success and subsequent development of rankbased inference, has been extended to many location problems, including problems involving multivariate and/or dependent observations. In this paper, we show that this uniform dominance also holds in problems for which the parameter of interest is the shape of an elliptical distribution. The Pitman nonadmissibility of the pseudoGaussian maximum likelihood estimator for shape and that of the pseudoGaussian likehood ratio test of sphericity follow.
Optimal detection of Fechnerasymmetry
 J. Statist. Plann. Inference
, 2008
"... We consider a general class of skewed univariate densities introduced by Fechner (1897), and derive optimal testing procedures for the null hypothesis of symmetry within that class. Locally and asymptotically optimal (in the Le Cam sense) tests are obtained, both for the case of symmetry with respec ..."
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Cited by 8 (2 self)
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We consider a general class of skewed univariate densities introduced by Fechner (1897), and derive optimal testing procedures for the null hypothesis of symmetry within that class. Locally and asymptotically optimal (in the Le Cam sense) tests are obtained, both for the case of symmetry with respect to a specified location as for the case of symmetry with respect to some unspecified location. Signedrank based versions of these tests are also provided. The efficiency properties of the proposed procedures are investigated by a derivation of their asymptotic relative efficiencies with respect to the corresponding Gaussian parametric tests based on the traditional PearsonFisher coefficient of skewness. Smallsample performances under several types of asymmetry are investigated via simulations.
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 8 (5 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
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 REstimation of a Spherical Location
, 2011
"... In this paper, we provide Restimators of the location of a rotationally symmetric distribution on the unit sphere of R k. In order to do so we first prove the local asymptotic normality property of a sequence of rotationally symmetric models; this is a non standard result due to the curved nature ..."
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Cited by 8 (8 self)
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In this paper, we provide Restimators of the location of a rotationally symmetric distribution on the unit sphere of R k. In order to do so we first prove the local asymptotic normality property of a sequence of rotationally symmetric models; this is a non standard result due to the curved nature of the unit sphere. We then construct our estimators by adapting the Le Cam onestep methodology to spherical statistics and ranks. We show that they are asymptotically normal under any rotationally symmetric distribution and achieve the efficiency bound under a specific density. Their small sample behavior is studied via a Monte Carlo simulation and our methodology is illustrated on geological data.
Supplement to “Semiparametrically efficient inference based on signed ranks in symmetric independent component models.” DOI:10.1214/11AOS906SUPP
, 2011
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Supplement to “Universal asymptotics for highdimensional sign tests
, 2013
"... In a smalln largep hypothesis testing framework, most procedures in the literature require quite stringent distributional assumptions, and restrict to a specific scheme of (n, p)asymptotics. More precisely, multinormality is almost always assumed, and it is imposed, typically, that p/n → c, fo ..."
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Cited by 5 (3 self)
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In a smalln largep hypothesis testing framework, most procedures in the literature require quite stringent distributional assumptions, and restrict to a specific scheme of (n, p)asymptotics. More precisely, multinormality is almost always assumed, and it is imposed, typically, that p/n → c, for some c in some given convex set C ⊂ (0,∞). Such restrictions clearly jeopardize practical relevance of these procedures. In this paper, we consider several classical testing problems in multivariate analysis, directional statistics, and multivariate time series: the problem of testing uniformity on the unit sphere, the spherical location problem, the problem of testing that a process is white noise versus serial dependence, the problem of testing for multivariate independence, and the problem of testing for sphericity. In each case, we show that the natural sign tests enjoy nonparametric validity and are distributionfree in a “universal ” (n, p)asymptotics framework, where p may go to infinity in an arbitrary way as n does. Simulations confirm our asymptotic results. 1. Introduction. There
Le Cam optimal tests for symmetry against Ferreira and Steel’s general skewed distributions
 J. Nonparam. Statist
, 2009
"... When testing symmetry of a univariate density, (parametric classes of) densities skewed by means of the general probability transform introduced in ..."
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Cited by 5 (3 self)
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When testing symmetry of a univariate density, (parametric classes of) densities skewed by means of the general probability transform introduced in