Proximal Estimation and Inference

Abstract: We build a unifying convex analysis framework characterizing the statistical properties of a large class of penalized estimators, both under a regular and an irregular design. Our framework interprets penalized estimators as proximal estimators, defined by a proximal operator applied to a corresponding initial estimator. We characterize 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's subgradients. We exploit our approach to systematically cover linear regression settings with a regular or irregular design. For these settings, we build new $\sqrt{n}-$consistent, asymptotically normal Ridgeless-type proximal estimators, which feature the Oracle property and are shown to perform satisfactorily in practically relevant Monte Carlo settings.