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A0799
Title: Proximal estimation and inference Authors:  Alberto Quaini - Columbia University (United States) [presenting]
Fabio Trojani - University of Geneva, University of Turin and SFI (Switzerland)
Abstract: A unifying convex analysis framework characterizing the statistical properties of a large class of penalized estimators is built, both under a regular or irregular design. The framework interprets penalized estimators as proximal estimators, defined by a proximal operator applied to a corresponding initial estimator. We obtain new characterizations of the asymptotic properties of proximal estimators, showing that their asymptotic distribution follows a closed-form formula depending only on (i) the asymptotic distribution of the initial estimator, (ii) the estimator's limit penalty subgradient and (iii) the inner product defining the associated proximal operator. In parallel, we characterize the Oracle features of proximal estimators from the properties of their penalty subgradients. We exploit our approach to systematically cover linear regression settings with a regular, singular or nearly singular design. For these settings, we build new root-$n$ consistent, asymptotically normal Ridgeless-type proximal estimators, which feature the Oracle property and are shown to perform satisfactorily in practically relevant Monte Carlo settings.