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28
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,
RANKBASED OPTIMAL TESTS OF THE ADEQUACY OF AN ELLIPTIC VARMA MODEL
, 2002
"... We are deriving optimal rankbased tests for the adequacy of a vector autoregressivemoving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudoMahalanobis distances and on normed residuals computed from Tyler’s [Ann. Statist. 15 (1987) ..."
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Cited by 22 (18 self)
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We are deriving optimal rankbased tests for the adequacy of a vector autoregressivemoving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudoMahalanobis distances and on normed residuals computed from Tyler’s [Ann. Statist. 15 (1987) 234–251] scatter matrix; they generalize the univariate signed rank procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1–29]. Two types of optimality properties are considered, both in the local and asymptotic sense, a la Le Cam: (a) (fixedscore procedures) local asymptotic minimaxity at selected radial densities, and (b) (estimatedscore procedures) local asymptotic minimaxity uniform over a class F of radial densities. Contrary to their classical counterparts, based on crosscovariance matrices, these tests remain valid under arbitrary elliptically symmetric innovation densities, including those with infinite variance and heavytails. We show that the AREs of our fixedscore
Equivariance and Invariance Properties of Multivariate Quantile and Related Functions, and the Role of Standardization
, 2009
"... Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invar ..."
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Cited by 17 (7 self)
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Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invariance of procedures, three important types of matrixvalued functional are studied: “weak covariance ” (or “shape”), “transformationretransformation ” (TR), and “strong invariant coordinate system ” (SICS). Clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artifacts of SICSstandardized data are invariant under affine transformation of the original data followed by standardization using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardization are described.
2002c). Optimal procedures based on interdirections and pseudoMahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence
 Bernoulli
"... We propose a multivariate generalization of signedrank tests for testing elliptically symmetric white noise against ARMA serial dependence. These tests are based on Randles (1989)’s concept of interdirections and the ranks of pseudoMahalanobis distances. They are affineinvariant, and asymptotica ..."
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Cited by 13 (10 self)
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We propose a multivariate generalization of signedrank tests for testing elliptically symmetric white noise against ARMA serial dependence. These tests are based on Randles (1989)’s concept of interdirections and the ranks of pseudoMahalanobis distances. They are affineinvariant, and asymptotically equivalent to strictly distributionfree statistics. Depending on the score function considered (van der Waerden, Laplace,...), they allow for locally asymptotically maximin tests at selected densities (multivariate normal, multivariate doubleexponential,...). Local powers and asymptotic relative efficiencies with respect to the Gaussian procedure are derived. We extend to the multivariate serial context the ChernoffSavage result, showing that classical correlogrambased procedures are uniformly dominated by the van der Waerden version of our tests, so that correlogram methods are not admissible in the Pitman sense. We also prove an extension of the celebrated HodgesLehmann “.864 result”, providing, for any fixed space dimension, the lower bound for the asymptotic relative efficiency of the proposed multivariate Spearman type tests with respect to the Gaussian tests. These asymptotic results are confirmed by a MonteCarlo investigation.
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 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