Orateur : Patrick Tardivel (IMB)

Résumé

Première partie (25 minutes environ) :   

Titre : Powerful multiple testing procedures derived from hyperrectangular confidence regions having a minimal volume

Résumé : We study the control of the FamilyWise Error Rate (FWER) in the linear Gaussian model when the nxp design matrix is of rank p. Single step multiple testing procedures controlling the FWER are derived from hyperrect- angular confidence regions. In this study, we aim to construct procedure derived from hyperrectangular confidence region having a minimal volume. We show that minimizing the volume seems a fair criterion to improve the power of the multiple testing procedure. Numerical experiments demonstrate the performance of our approach when compared with the state-of-the-art single step and sequential procedures. We also provide an application to the detection of metabolites in metabolomics.

Deuxième partie (25 minutes environ)

Titre : The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation

Résumé :  We provide a necessary and sufficient condition for the uniqueness of penalized least-squares estimators whose penalty term is given by a norm with a polytope unit ball, covering a wide range of methods including SLOPE and LASSO, as well as the related method of basis pursuit. We consider a strong type of uniqueness that is relevant for statistical problems. The uniqueness condition is geometric and involves how the row span of the design matrix intersects the faces of the dual norm unit ball, which for SLOPE is given by the sign permutahedron. Further considerations based on this condition also allow to derive results on sparsity and clustering features. In particular, we define the notion of a SLOPE model to describe both sparsity and clustering properties of this method and also provide a geometric characterization of accessible SLOPE models.

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