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

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.

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-

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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 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.

Document de travail

mixtures of beta densities in estimating positive false discovery rates

We propose a class of rank-based 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 distribution-free 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

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. Signed-rank 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 Pearson-Fisher coefficient of skewness. Small-sample performances under several types of asymmetry are investigated via simulations.