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22
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 47 (31 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 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
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 signedrank tests based on hyperplanes
 hal00655850, version 2  16
, 2005
"... For analysing kvariate data sets, Randles (1989) considered hyperplanes going through k − 1 data points and the origin. He then introduced an empirical angular distance between two kvariate data vectors based on the number of hyperplanes (the socalled interdirections) that separate these two poi ..."
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Cited by 8 (3 self)
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For analysing kvariate data sets, Randles (1989) considered hyperplanes going through k − 1 data points and the origin. He then introduced an empirical angular distance between two kvariate data vectors based on the number of hyperplanes (the socalled interdirections) that separate these two points, and proposed a multivariate sign test based on those interdirections. In this paper, we present an analogous concept (namely, liftinterdirections) to measure the regular distances between data points. The empirical distance between two kvariate data vectors is again determined by the number of hyperplanes that separate these two points; in this case, however, the considered hyperplanes are going through k distinct data points. The invariance and convergence properties of the empirical distances are considered. We show that the liftinterdirections together with Randles ’ interdirections allow for building hyperplanebased versions of the optimal testing procedures developed in Hallin and Paindaveine (2002a, b, c, and 2004a) for a broad class of location and time series problems. The resulting procedures, which generalize the univariate signedrank procedures, are affineinvariant and asymptotically invariant under a group of monotone radial transformations (acting on the standardized residuals). Consequently, they are asymptotically distributionfree under the class of elliptical distributions. They are optimal under correctly specified radial densities and, in several cases, enjoy a uniformly good efficiency behavior. These asymptotic properties are confirmed by a MonteCarlo study, and, finally, a simple robustness study is conducted. It is remarkable that, in the test construction, the value of the test statistic depends on the data cloud only through the geometrical notions of data vectors and oriented hyperplanes, and their relations “above ” and “below”.
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 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.
Multivariate signed ranks : Randles’ interdirections or Tyler’s angles
 In Statistical data analysis based on the L1norm and related methods
"... Abstract. Hallin and Paindaveine (2002a) developed, for the multivariate (elliptically symmetric) onesample location problem, a class of optimal procedures, based on Randles ’ interdirections and the ranks of pseudoMahalanobis distances. We present an alternative version of these procedures in whi ..."
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Cited by 6 (4 self)
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Abstract. Hallin and Paindaveine (2002a) developed, for the multivariate (elliptically symmetric) onesample location problem, a class of optimal procedures, based on Randles ’ interdirections and the ranks of pseudoMahalanobis distances. We present an alternative version of these procedures in which interdirections are replaced by “Tyler angles”, namely, the angles between the observations standardized via Tyler’s estimator of scatter. These Tyler angles are indeed computationally preferable (in terms of CPU time) to interdirections. We show that the two approaches are asymptotically equivalent. A MonteCarlo study is conducted to compare their smallsample efficiency and robustness features. Simulations indicate that, whereas interdirections and Tyler angles yield comparable results under strict ellipticity and radial outliers, interdirections are significantly more reliable in the presence of angular outliers. This study is focused on the simple onesample location problem. It readily extends, with obvious changes, to more complex models such as multivariate regression or analysis of variance, and to time series models (see
Affine invariant linear hypotheses for the multivariate general linear model with VARMA error terms
 In Mathematical Statistics and Applications: Festschrift for Constance
, 2003
"... Affine invariance is often considered a natural requirement when testing hypotheses in a multivariate context. This invariance issue is considered here in the problem of testing linear constraints on the parameters of a multivariate linear model with VARMA error terms. We give a characterization of ..."
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Cited by 4 (2 self)
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Affine invariance is often considered a natural requirement when testing hypotheses in a multivariate context. This invariance issue is considered here in the problem of testing linear constraints on the parameters of a multivariate linear model with VARMA error terms. We give a characterization of the collection of null hypotheses that are invariant under the group of affine transformations, hence compatible with a requirement of affine invariant testing. We comment the results and discuss some examples. 1 Introduction. Affine invariance/equivariance often is considered a natural requirement in multivariate statistical inference. The rationale for such a requirement is that the data at hand, or the noise underlying the model, should be treated as intrinsically multivariate objects, irrespective of any particular choice of a coordinate system. This requirement plays a fundamental role in most
A unified and elementary proof of serial and nonserial, univariate and multivariate, ChernoffSavage results
 Statist. Methodol
, 2004
"... We provide a simple proof that the ChernoffSavage [1] result, establishing the uniform dominance of normalscore rank procedures over their Gaussian competitors, also holds in a broad class of problems involving serial and/or multivariate observations. The nonadmissibility of the corresponding e ..."
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Cited by 2 (2 self)
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We provide a simple proof that the ChernoffSavage [1] result, establishing the uniform dominance of normalscore rank procedures over their Gaussian competitors, also holds in a broad class of problems involving serial and/or multivariate observations. The nonadmissibility of the corresponding everyday practice Gaussian procedures (multivariate leastsquares estimators, multivariate ttests and Ftests, correlogrambased methods, multivariate portmanteau and DurbinWatson tests, etc.) follows. The proof, which generalizes to the multivariate—possibly serial—setup the idea developed in Gastwirth and Wolff [2] in the context of univariate location problems, allows for avoiding technical convexity and variational arguments. Key words: Pitmaninadmissibility, rankbased inference, ChernoffSavage results, multivariate signs and ranks.
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.