Davy Paindaveine

Semiparametrically efficient rank-based inference for shape I. Optimal rank-based tests for sphericity

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 V 0 ; this includes, for V 0 = I k , 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 must be estimated. The multivariate ranks used throughout are those of the distances-in the metric associated with the null value V 0 of the shape matrix-between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version of the Gaussian likelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67 (1980) 31-43]) or, equivalently, with respect to (an extension of) the test for sphericity introduced by John [Biometrika 59 (1972) 169-173]. For Gaussian scores, these AREs are uniformly larger than one, irrespective of the actual radial density. Necessary and/or sufficient conditions for consistency under nonlocal, possibly nonelliptical alternatives are given. Finite sample performance is investigated via a Monte Carlo study.

Semiparametrically efficient rank-based inference for shape. II. Optimal R-estimation of shape

A class of R-estimators, based on the concepts of multivariate signed ranks and the optimal rank-based tests developed in Hallin and Paindaveine (2006a), is proposed for the estimation of the shape matrix of an elliptical distribution. These R-estimators are root-n 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 thanthe traditional normal-theory 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 rank-based version of Le Cam’s one-step methodology, which avoids the unpleasant nonparametric estimation of cross-information quantities that is generally required in the context of R-estimation. Although they are not strictly equivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasability and excellent finite-sample performances.

Computing multiple-output regression quantile regions

A procedure relying on linear programming techniques is developed to compute (regression) quantile regions that have been defined recently. In the location case, this procedure allows for computing halfspace depth regions even beyond dimension two. The corresponding algorithm is described in detail, and illustrations are provided both for simulated and real data. The efficiency of a Matlab implementation of the algorithm is also investigated through extensive simulations.

From depth to local depth: : A focus on centrality

Aiming at analyzing multimodal or nonconvexly supported distributions through data depth, we introduce a local extension of depth. Our construction is obtained by conditioning the distribution to appropriate depth-based neighborhoods and has the advantages, among others, of maintaining affine-invariance and applying to all depths in a generic way. Most importantly, unlike their competitors, which (for extreme localization) rather measure probability mass, the resulting local depths focus on centrality and remain of a genuine depth nature at any locality level. We derive their main properties, establish consistency of their sample versions, and study their behavior under extreme localization. We present two applications of the proposed local depth (for classification and for symmetry testing), and we extend our construction to the regression depth context. Throughout, we illustrate the results on several datasets, both artificial and real, univariate and multivariate. Supplementary materials for this article are available online.

On directional multiple-output quantile regression

This paper sheds some new light on projection quantiles. Contrary to the sophisticated set analysis used in Kong and Mizera (2008) [13], we adopt a more parametric approach and study the subgradient conditions associated with these quantiles. In this setup, we introduce Lagrange multipliers which can be interpreted in various interesting ways, in particular in a portfolio optimization context. The corresponding projection quantile regions were already shown to coincide with the halfspace depth ones in Kong and Mizera (2008) [13], but we provide here an alternative proof (completely based on projection quantiles) that has the advantage of leading to an exact computation of halfspace depth regions from projection quantiles. Above all, we systematically consider the regression case, which was barely touched in Kong and Mizera (2008) [13]. We show in particular that the regression quantile regions introduced in Hallin, Paindaveine, and Siman (2010) [6,7] can also be obtained from projection (regression) quantiles, which may lead to a faster computation of those regions in some particular cases.

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985)—and their multivariate generalizations from Azzalini and Dalla Valle (1996)—have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti 2004, 2009) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew- exponential or skew-t distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? In this talk, we provide an answer to this question. In very general (possibly multivariate) skew-symmetric models (see Ma and Genton 2004), we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference. References: A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist 12 (1985) 171-178. A. Azzalini, A. Dalla Valle, The multivariate skew-normal distribution, Biometrika 83 (1996) 715-726. T.J. DiCiccio, A.C. Monti, Inferential aspects of the skew exponential power distribution, J. Amer. Statist. Assoc. 99 (2004) 439-450. T.J. DiCiccio, A.C. Monti, Inferential aspects of the skew t- distribution (2009). Manuscript in preparation. Y. Ma, M.G. Genton, Flexible class of skew-symmetric distributions, Scand. J. Statist. 31 (2004) 459-468.

Optimal rank-based testing for principal components

This paper provides parametric and rank-based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudo-Gaussian robustifications by Tyler (1981, 1983) and Davis (1977), with which their Gaussian versions are shown to coincide,symptotically, under Gaussian or finite fourth-order moment assumptions, respectively. Such assumptions however restrict the scope to covariance-based principal component analysis. The rank-based tests we are proposing remain valid without such assumptions. Hence, they address a much broader class of problems, where covariance matrices need not exist and principal components are associated with more general scatter matrices. Asymptotic relative efficiencies moreover show that those rank-based tests are quite powerful; when based on van der Waerden or normal scores, they even uniformly dominate the pseudo-Gaussian versions of Anderson’s procedures. The tests we are proposing thus outperform daily practice both from the point of view of validity as from the point of view of efficiency. The main methodological tool throughout is Le Cam’s theory of locally asymptotically normal experiments, in the nonstandard context, however, of a curved parametrization. The results we derive for curved experiments are of independent interest,and likely to apply in other setups.

Rank-based optimal tests of the adequacy of an elliptic VARMA model

We are deriving optimal rank-based tests for the adequacy of a vector autoregressive-moving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudo-Mahalanobis distances and on narmed residuals computed from Tylers [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) (fixed-score procedures) local asymptotic minimaxity at selected radial densities, and (b) (estimated-score procedures) local asymptotic minimaxity uniform over a class F of radial densities. Contrary to their classical counterparts, based on cross-covariance matrices, these tests remain valid under arbitrary elliptically symmetric innovation densities, including those with infinite variance and heavy-tails. We show that the AREs of our fixed-score procedures, with respect to traditional (Gaussian) methods, are the same as for the tests of randomness proposed in Hallin and Paindaveine [Bernoulli 8 (2002b) 787-815]. The multivariate serial extensions of the classical Chernoff-Savage and Hodges-Lehmann results obtained there thus also hold here; in particular, the van der Waerden versions of our tests are uniformly more powerful than those based on cross-covariances. As for our estimated-score procedures, they are fully adaptive, hence, uniformly optimal over the class of innovation densities satisfying the required technical assomptions.

OPTIMAL RANK-BASED 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 perfectly robust against heavy-tailed 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 normal-score version outperforms traditional Gaussian likelihood ratio tests and their pseudo-Gaussian robustifications under a very broad range of non-Gaussian densities including, for instance, all multivariate Student and power-exponential distributions.

Semiparametrically efficient inference based on signed ranks in symmetric independent component models

We consider semiparametric location-scatter models for which the p-variate observation is obtained as X=Lambda Z+mu, where mu is a p-vector, Lambda is a full-rank p*p matrix, and the (unobserved) random p-vector Z has marginals that are centered and mutually independent but are otherwise unspecified. As in blind source separation and independent component analysis (ICA), the parameter of interest throughout the paper is Lambda. On the basis of n i.i.d. copies of X, we develop, under a symmetry assumption on Z, signed-rank one-sample testing and estimation procedures for Lambda. We exploit the uniform local and asymptotic normality (ULAN) of the model to define signed-rank procedures that are semiparametrically efficient under correctly specified densities. Yet, as usual in rank-based inference, the proposed procedures remain valid (correct asymptotic size under the null, for hypothesis testing, and root-n consistency, for point estimation) under a very broad range of densities. We derive the asymptotic properties of the proposed procedures and investigate their finite-sample behavior through simulations.