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14
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,
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 21 (17 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
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 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”.
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.
IMS Collections Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor
"... mixtures of beta densities in estimating positive false discovery rates ..."