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Optimal rankbased tests for common principal components
 Bernoulli
, 2013
"... This paper provides optimal testing procedures for the msample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous elliptical densities. We first establish, under very mild assumptions that do not require finite moments of order four, the local asympto ..."
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This paper provides optimal testing procedures for the msample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous elliptical densities. We first establish, under very mild assumptions that do not require finite moments of order four, the local asymptotic normality (LAN) of the model. Based on that result, we show that the pseudoGaussian test proposed in Hallin et al. (2010a) is locally and asymptotically optimal under Gaussian densities. We also show how to compute its local powers and asymptotic relative efficiencies (AREs). A numerical evaluation of those AREs, however, reveals that, while remaining valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rankbased procedures that remain valid under any possibly heterogenousmtuple of elliptical densities, irrespective of any moment assumptions—in elliptical families, indeed, principal components naturally can be based on the scatter matrices characterizing the density contours, hence do not require finite variances. Those rankbased tests are not only validityrobust in the sense that they survive arbitrary elliptical population ∗Académie Royale de Belgique, CentER, Tilburg University, and ECORE. Research supported by the Sonderforschungsbereich “Statistical modelling of nonlinear dynamic processes ” (SFB 823) of the
PARAMETRIC AND NONPARAMETRIC TESTS FOR MULTIVARIATE INDEPENDENCE IN IC MODELS
"... The socalled independent component (IC) model states that the observed pvector X is generated via X = ΛZ + µ, where µ is a pvector, Λ is an invertible matrix, and the centered random vector Z has independent marginals Zi. We consider the problem of testing, on the basis of n i.i.d. copies of X = ..."
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The socalled independent component (IC) model states that the observed pvector X is generated via X = ΛZ + µ, where µ is a pvector, Λ is an invertible matrix, and the centered random vector Z has independent marginals Zi. We consider the problem of testing, on the basis of n i.i.d. copies of X = (X (1) ′ , X (2) ′ ) ′ , the null hypothesis under which the multivariate marginals X (1) and X (2) are independent. Under a symmetry assumption on the Zi’s, we propose parametric and nonparametric tests based on estimated independent components (which are obtained under the null, via, e.g., a recent estimator due to Oja et al. 2006). Far from excluding cases of unidentifiability where several independent components are Gaussian, as it is done in the socalled independent component analysis (ICA), our procedures can deal with the resulting possible model singularity, the nature of which we carefully investigate. The proposed nonparametric tests are based on componentwise signed ranks, in the same spirit as in Puri and Sen (1971). However, unlike the Puri and Sen tests, our tests (i) are affineinvariant and (ii) are, for adequately chosen scores, locally and asymptotically optimal (in the Le Cam sense) at prespecified densities. They are also valid without any moment assumptions. Local powers and asymptotic relative efficiencies with respect to the classical Gaussian procedure (namely, Wilks ’ LRT) are derived. Finitesample properties are investigated through a MonteCarlo study.
Restimation for asymmetric independent component analysis
 Journal of the American Statistical Association
, 2014
"... Independent Component Analysis (ICA) recently has attracted much attention in the statistical literature as an appealing alternative to elliptical models. Whereas kdimensional elliptical densities depend on one single unspecified radial density, however, kdimensional independent component distribu ..."
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Independent Component Analysis (ICA) recently has attracted much attention in the statistical literature as an appealing alternative to elliptical models. Whereas kdimensional elliptical densities depend on one single unspecified radial density, however, kdimensional independent component distributions involve k unspecified component densities. In practice, for given sample size n and dimension k, this makes the statistical analysis much harder. We focus here on the estimation, from an independent sample, of the mixing/demixing matrix of the model. Traditional methods (FOBI, KernelICA, FastICA) mainly originate from the engineering literature. Their consistency requires moment conditions, they are poorly robust, and do not achieve any type of asymptotic efficiency. When based on robust scatter matrices, the twoscatter methods developed by Oja et al. (2006) and Nordhausen et al. (2008) enjoy better robustness features, but their optimality properties remain unclear. The “classical semiparametric ” approach by Chen and Bickel (2006), quite on the contrary, achieves semiparametric efficiency, but requires the estimation of the densities of the k unobserved independent compo
Semiparametrically efficient inference based on signs and ranks for medianrestricted models
, 2008
"... ..."
A class of optimal tests for symmetry based on local edgeworth approximations
 Bernoulli
, 2011
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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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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 ..."
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
Davy PAINDAVEINEOptimal Detection of FechnerAsymmetry
, 2007
"... We consider a general class of skewed univariate densities introduced by Fechner (1897), and derive optimal testing procedures for the null hypothesis of symmetry within that class. Locally and asymptotically optimal (in the Le Cam sense) tests are obtained, both for the case of symmetry with respec ..."
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We consider a general class of skewed univariate densities introduced by Fechner (1897), and derive optimal testing procedures for the null hypothesis of symmetry within that class. Locally and asymptotically optimal (in the Le Cam sense) tests are obtained, both for the case of symmetry with respect to a specified location as for the case of symmetry with respect to some unspecified location. Signedrank based versions of these tests are also provided. The efficiency properties of the proposed procedures are investigated by a derivation of their asymptotic relative efficiencies with respect to the corresponding Gaussian parametric tests based on the traditional PearsonFisher coefficient of skewness. Smallsample performances under several types of asymmetry are investigated via simulations.