Proximal Estimation and Inference

We develop a convex analysis framework to construct and study with a single approach a new broad class of penalized estimators. Our method builds penalized estimators by applying proximal operators to suitable initial estimators. This construction enables us to systematically determine the asymptotic properties of the resulting proximal estimators. We provide general closed-form expressions for their asymptotic distributions, which depend on three key components: (i) the asymptotic distribution of the initial estimator, (ii) the subgradient of the limit penalty and (iii) the inner product defining the associated proximal operator. We further obtain general characterizations of their Oracle properties, which depend exclusively on their penalty's subgradient. We apply our framework to linear regression models with a potential design nearly-singularity, in order to build new Ridgeless-type proximal estimators, which are root-n-consistent, asymptotically normal, and giving rise to Oracle proximal estimators.