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.
