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22
Optimal Sparse Principal Component Analysis in High Dimensional Elliptical Model
, 2013
"... We propose a semiparametric sparse principal component analysis method named elliptical component analysis (ECA) for analyzing high dimensional nonGaussian data. In particular, we assume the data follow an elliptical distribution. Elliptical family contains many wellknown multivariate distributio ..."
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We propose a semiparametric sparse principal component analysis method named elliptical component analysis (ECA) for analyzing high dimensional nonGaussian data. In particular, we assume the data follow an elliptical distribution. Elliptical family contains many wellknown multivariate distributions such as multivariate Gaussian, multivariatet, Cauchy, Kotz, and logistic distributions. It allows extra flexibility on modeling heavytailed distributions and capture tail dependence between variables. Such modeling flexibility makes it extremely useful in modeling financial, genomics and bioimaging data, where the data typically present heavy tails and high tail dependence. Under a double asymptotic framework where both the sample size n and the dimension d increase, we show that a multivariate rank based ECA procedure attains the optimal rate of convergence in parameter estimation. This is the first optimality result established for sparse principal component analysis on high dimensional elliptical data.
The Annals of Statistics SEMIPARAMETRICALLY EFFICIENT RANKBASED INFERENCE FOR SHAPE I. OPTIMAL RANKBASED TESTS FOR SPHERICITY
"... We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. T ..."
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We propose a class of rankbased procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations, and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distributionfree when the center of symmetry is specified, and asymptotically so, when it has to be estimated. The multivariate ranks used throughout are those of the distances—in the metric associated with the null value V0 of the shape matrix—between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and
Robust Optimal Tests for Causality in Multivariate Time Series ∗
"... Here, we derive optimal rankbased tests for noncausality in the sense of Granger between two multivariate time series. Assuming that the global process admits a joint stationary vector autoregressive (VAR) representation with an elliptically symmetric innovation density, both no feedback and one di ..."
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Here, we derive optimal rankbased tests for noncausality in the sense of Granger between two multivariate time series. Assuming that the global process admits a joint stationary vector autoregressive (VAR) representation with an elliptically symmetric innovation density, both no feedback and one direction causality hypotheses are tested. Using the characterization of noncausality in the VAR context, the local asymptotic normality (LAN) theory described in Le Cam (1986)) allows for constructing locally and asymptotically optimal tests for the null hypothesis of noncausality in one or both directions. These tests are based on multivariate residual ranks and signs (Hallin and Paindaveine, 2004a) and are shown to be asymptotically distribution free under elliptically symmetric innovation densities and invariant with respect to some affine transformations. Local powers and asymptotic relative efficiencies are also derived. The level, power and robustness (to outliers) of the resulting tests are studied by simulation and are compared to those of Wald test. Finally, the new tests are applied to Canadian money and income data.
Multivariate SignedRank Tests in Vector Autoregressive Order Identification
"... Abstract. The classical theory of rankbased inference is essentially limited to univariate linear models with independent observations. The objective of this paper is to illustrate some recent extensions of this theory to timeseries problems (serially dependent observations) in a multivariate sett ..."
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Abstract. The classical theory of rankbased inference is essentially limited to univariate linear models with independent observations. The objective of this paper is to illustrate some recent extensions of this theory to timeseries problems (serially dependent observations) in a multivariate setting (multivariate observations) under very mild distributional assumptions (mainly, elliptical symmetry; for some of the testing problems treated below, even secondorder moments are not required). After a brief presentation of the invariance principles that underlie the concepts of ranks to be considered, we concentrate on two examples of practical relevance: (1) the multivariate Durbin–Watson problem (testing against autocorrelated noise in a linear model context) and (2) the problem of testing the order of a vector autoregressive model, testing VAR(p0) against VAR(p0 + 1) dependence. These two testing procedures are the building blocks of classical autoregressive orderidentification methods. Based either on pseudoMahalanobis (Tyler) or on hyperplanebased (Oja and Paindaveine) signs and ranks, three classes of test statistics are considered for each problem: (1) statistics of the signtest type, (2) Spearman statistics and (3) van der Waerden (normal score) statistics. Simulations confirm theoretical results about the power of the proposed rankbased methods and establish their good robustness properties.
(1)Institut de Statistique et de Recherche Opérationnelle Campus de la Plaine CP 210 Universite ́ libre de Bruxelles B1050 Bruxelles
"... Nous proposons un Restimateur de la matrice de forme d’une loi elliptique de densite ́ radiale et de paramètre d’échelle nonspécifiés. Notre estimateur prend la forme d’un estimateur “onestep”, et s’écrit comme une combinaison linéaire de l’estimateur de Tyler V (n) T et d’une matrice de fo ..."
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Nous proposons un Restimateur de la matrice de forme d’une loi elliptique de densite ́ radiale et de paramètre d’échelle nonspécifiés. Notre estimateur prend la forme d’un estimateur “onestep”, et s’écrit comme une combinaison linéaire de l’estimateur de Tyler V (n) T et d’une matrice de forme mesurable en les rangs et signes multivariés des observations. Ces rangs et signes sont soit les rangs des distances, calculées dans la métrique associée a ̀ V (n) T, entre les observations et le centre (éventuellement estimé) de la loi, soit les rangs de OjaPaindaveine, obtenus sur la base de comptages d’hyperplans séparateurs. Notre estimateur est nconvergent et asymptotiquement normal en l’absence de toute hypothèse sur les moments de la loi sousjacente. Il est asymptotiquement équivariant par transformation affine. Pour un choix de scores adéquat, il est localement et asymptotiquement minimax (au sens de Le Cam). Fonde ́ sur des scores estimés, il est uniformément minimaxefficace, au sens semiparamétrique du terme, sur une classe très générale de densités radiales.
2013/30 On Hodges and Lehman's "6/pi Result"
"... While the asymptotic relative efficiency (ARE) of Wilcoxon rankbased tests for location and regression with respect to their parametric Student competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that the ARE of the same Wilcoxon tests with respect to their van der Waerden or ..."
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While the asymptotic relative efficiency (ARE) of Wilcoxon rankbased tests for location and regression with respect to their parametric Student competitors can be arbitrarily large, Hodges and Lehmann (1961) have shown that the ARE of the same Wilcoxon tests with respect to their van der Waerden or normalscore counterparts is bounded from above by 6/π ≈ 1.910. In this paper, we revisit that result, and investigate similar bounds for statistics based on Student scores. We also consider the serial version of this ARE. More precisely, we study the ARE, under various densities, of the SpearmanWaldWolfowitz and Kendall rankbased autocorrelations with respect to the van der Waerden or normalscore ones used to test (ARMA) serial dependence alternatives.
The Annals of Statistics OPTIMAL RANKBASED TESTS FOR HOMOGENEITY OF SCATTER
"... 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 perfectl ..."
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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. ∗The authors are also members of ECORE, the recently created association between CORE and ECARES.
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