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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 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 Multivariate Runs Tests for Randomness
"... matrix This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999 ..."
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Cited by 1 (1 self)
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matrix This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and (ii) an original concept of matrixvalued runs. The resulting runs tests themselves exist in various versions, either based on spatial signs (see, e.g., Möttönen and Oja 1995, Randles 2000) or on the hyperplanebased multivariate signs known as interdirections (see, e.g., Randles 1989, Taskinen, Oja, and Randles 2005). All proposed multivariate runs tests are affineinvariant and highly robust: in particular, they allow for heteroskedasticity and do not require any moment assumption. Their limiting distributions are derived under the null hypothesis and under sequences of local vector ARMA alternatives. Asymptotic relative efficiencies with respect to Gaussian Portmanteau tests are computed, and show that, while Mardentype runs tests suffer severe consistency problems, tests based on matrixvalued runs perform uniformly well for moderatetolarge dimensions. A MonteCarlo study confirms the theoretical results and investigates the robustness properties of the proposed procedures. A real data example is treated, and shows that combining Mardentype runs tests and tests based on matrixvalued runs may provide some insight on the reason why rejection occurs.
Journal de la Société Française de Statistique Rank Tests for Elliptical Graphical Modeling Titre: Tests de Rangs pour les Modèles Graphiques Elliptiques
"... Abstract: As a reaction to the restrictive Gaussian assumptions that are usually part of graphical models, Vogel and Fried [17] recently introduced elliptical graphical models, in which the vector of variables at hand is assumed to have an elliptical distribution. The present work introduces a class ..."
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Abstract: As a reaction to the restrictive Gaussian assumptions that are usually part of graphical models, Vogel and Fried [17] recently introduced elliptical graphical models, in which the vector of variables at hand is assumed to have an elliptical distribution. The present work introduces a class of rank tests in the context of elliptical graphical models. The proposed tests are valid under any elliptical density, and in particular do not require any moment assumption. They achieve local and asymptotic optimality under correctly specified densities. Their asymptotic properties are investigated both under the null and under sequences of local alternatives. Asymptotic relative efficiencies with respect to the corresponding pseudoGaussian competitors are derived, which allows to show that, when based on normal scores, the proposed rank tests uniformly dominate the pseudoGaussian tests in the Pitman sense. The asymptotic results are confirmed through a MonteCarlo study. Résumé: En réaction aux hypothèses gaussiennes restrictives qui accompagnent le plus souvent les modèles graphiques, Vogel et Fried [17] ont récemment introduit des modèles graphiques elliptiques, qui prévoient que les variables suivent conjointement une distribution elliptique. Le présent travail introduit une classe de tests de rangs dans le contexte de ces modèles graphiques elliptiques. Ces tests sont valides sous une densité elliptique quelconque, et en particulier ne requièrent aucune hypothèse de moment. Ils sont localement et asymptotiquement optimaux sous des densités correctement spécifiées. Leurs propriétés asymptotiques sont étudiées à la fois sous l’hypothèse nulle et
Davy PAINDAVEINEDepthbased Runs Tests for Multivariate Central Symmetry
"... McWilliams (1990) introduced a nonparametric procedure based on runs for the problem of testing univariate symmetry about the origin (equivalently, about an arbitrary specified center). His procedure first reorders the observations according to their absolute values, then rejects the null when the n ..."
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McWilliams (1990) introduced a nonparametric procedure based on runs for the problem of testing univariate symmetry about the origin (equivalently, about an arbitrary specified center). His procedure first reorders the observations according to their absolute values, then rejects the null when the number of runs in the resulting series of signs is too small. This test is universally consistent and enjoys nice robustness properties, but is unfortunately limited to the univariate setup. In this paper, we extend McWilliams ’ procedure into tests of central symmetry in any dimension. The proposed tests first reorder the observations according to their statistical depth in a symmetrized version of the sample, then reject the null when an original concept of simplicial runs in the resulting series of (spatial) signs is too small. Our tests are affineinvariant and have good robustness properties. In particular, they do not require any finite moment assumption. We derive their limiting null distribution, which establishes their asymptotic distributionfreeness. We study their finitesample properties through Monte Carlo experiments, and conclude with some final comments.
On Multivariate Runs Tests for Randomness Davy PAINDAVEINE On Multivariate Runs Tests for Randomness
"... This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and ..."
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This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and (ii) an original concept of matrixvalued runs. The resulting runs tests themselves exist in various versions, one of which is a function of the number of databased hyperplanes separating pairs of observations only. All proposed multivariate runs tests are affineinvariant and highly robust: in particular, they allow for heteroskedasticity and do not require any moment assumption. Their limiting distributions are derived under the null hypothesis and under sequences of local vector ARMA alternatives. Asymptotic relative efficiencies with respect to Gaussian Portmanteau tests are computed, and show that, while Mardentype runs tests suffer severe consistency problems, tests based on matrixvalued
MONGEKANTOROVICH DEPTH, QUANTILES, RANKS AND SIGNS
"... Abstract. We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on IRd and a reference distribution on the ddimensional unit ball. The new depth concept, called MongeKantorovich depth, spec ..."
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Abstract. We propose new concepts of statistical depth, multivariate quantiles, ranks and signs, based on canonical transportation maps between a distribution of interest on IRd and a reference distribution on the ddimensional unit ball. The new depth concept, called MongeKantorovich depth, specializes to halfspace depth in the case of elliptical distributions, but, for more general distributions, differs from the latter in the ability for its contours to account for non convex features of the distribution of interest. We propose empirical counterparts to the population versions of those MongeKantorovich depth contours, quantiles, ranks and signs, and show their consistency by establishing a uniform convergence property for empirical transport maps, which is of independent interest.