The so-called independent component (IC) model states that the observed p-vector X is generated via X = ΛZ + µ, where µ is a p-vector, 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 so-called 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 affine-invariant 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. Finite-sample properties are investigated through a Monte-Carlo study.

EQUIPPE-GREMARS, Université Lille Nord de France. Classical estimation techniques for linear models either are inconsistent, or perform rather poorly, under α-stable error densities; most of them are not even rate-optimal. In this paper, we propose an original one-step R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators remain root-n consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While parametric stable-likelihood estimation, due to the absence of a closed form for stable densities, is quite cumbersome, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values (α0, b0) of the tail index α and skewness parameter b, while preserving root-n consistency under any (α, b) as well as under usual light-tailed densities. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.

This paper provides optimal testing procedures for the m-sample null hypothesis of Common Principal Components (CPC) under possibly non Gaussian and heterogenous elliptical densi-ties. 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 pseudo-Gaussian 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 remain-ing valid, this test is poorly efficient away from the Gaussian. Moreover, it still requires finite moments of order four. We therefore propose rank-based procedures that remain valid under any possibly heterogenousm-tuple 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 rank-based tests are not only validity-robust 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

Link to publication Citation for published version (APA): Croux, C., Dehon, C., & Yadine, A. (2010). The K-Step Spatial Sign Covariance Matrix. (CentER Discussion Paper; Vol. 2010-41). Tilburg: Econometrics. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.? Users may download and print one copy of any publication from the public portal for the purpose of private study or research? You may not further distribute the material or use it for any profit-making activity or commercial gain? You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 09. mei. 2016

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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 k-dimensional elliptical densities depend on one single unspecified radial density, however, k-dimensional 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, Kernel-ICA, 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 two-scatter 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-

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 matrix-valued 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 hyperplane-based multivariate signs known as interdirections (see, e.g., Randles 1989, Taskinen, Oja, and Randles 2005). All proposed multivariate runs tests are affine-invariant 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 Marden-type runs tests suffer severe consistency problems, tests based on matrix-valued runs perform uniformly well for moderate-to-large dimensions. A Monte-Carlo study confirms the theoretical results and investigates the robustness properties of the proposed procedures. A real data example is treated, and shows that combining Marden-type runs tests and tests based on matrix-valued runs may provide some insight on the reason why rejection occurs.

Abstract This work is concerned with testing the population mean vector of nonnormal high-dimensional multivariate data. Several tests for high-dimensional mean vector have been proposed in the literature, but they may not perform well for high-dimensional continuous, nonnormal multivariate data, which frequently arise in genomics studies and quantitative finance. This paper aims to develop a novel high-dimensional nonparametric test for the population mean vector so that multivariate normality assumption becomes unnecessary. With the aid of new tools in modern probability theory, we proved that the limiting null distribution of the proposed test is normal under mild conditions for p > n. We further study the local power of the proposed test and compare its relative efficiency with a modified Hotelling T 2 test for high-dimensional data. Our theoretical results indicate that the newly proposed test can have even more substantial power gain than the traditional nonparametric multivariate test does with finite fixed p. We assess the finite sample performance of the proposed test by examining its size and power via Monte Carlo studies. We illustrate the application of the proposed test by an empirical analysis of a genomics data set.

We propose a semiparametric sparse principal component analysis method named el-liptical component analysis (ECA) for analyzing high dimensional non-Gaussian data. In particular, we assume the data follow an elliptical distribution. Elliptical family contains many well-known multivariate distributions such as multivariate Gaussian, multivariate-t, Cauchy, Kotz, and logistic distributions. It allows extra flexibility on modeling heavy-tailed 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 dimen-sion 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.

Abstract The objective of this paper is to provide, for the problem of univariate symmetry (with respect to specified or unspecified location), a concept of optimality, and to construct tests achieving such optimality. This requires embedding symmetry into adequate families of asymmetric (local) alternatives. We construct such families by considering non-Gaussian generalizations of classical first-order Edgeworth expansions indexed by a measure of skewness such that (i) location, scale and skewness play well-separated roles (diagonality of the corresponding information matrices), and (ii) the classical tests based on the Pearson-Fisher coefficient of skewness are optimal in the vicinity of Gaussian densities.