PROXIMAL ESTIMATION AND INFERENCE

Abstract We develop a convex-analytic framework for constructing and analyzing a broad class of penalized estimators. Our method defines penalized estimators by applying proximal operators to benchmark estimators, rather than by penalizing the benchmark criterion itself. We establish general finite-sample properties, derive unified asymptotic distribution results, and provide tractable conditions for oracle selection and oracle asymptotic efficiency. We then apply the framework to linear regression with irregular designs, including singular and nearly-singular cases. In this setting, where the coefficient vector need not be point identified, we motivate the Ridgeless estimand as the inferential target, construct a n $\sqrt {n}$ StartRoot n EndRoot -consistent modified Ridgeless benchmark estimator, and combine it with adaptive proximal penalization to obtain, to the best of our knowledge, the first penalized estimator achieving the oracle property under both regular and irregular designs.