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.

翻译：我们构建了一个统一的凸分析框架，用于刻画一大类惩罚估计量的统计性质，该框架同时适用于规则设计与不规则设计情形。我们的框架将惩罚估计量解释为邻近估计量，即通过将邻近算子作用于相应的初始估计量而定义。我们刻画了邻近估计量的渐近性质，证明其渐近分布遵循一个闭式公式，该公式仅依赖于：(i) 初始估计量的渐近分布，(ii) 估计量极限惩罚次梯度的性质，以及 (iii) 定义相关邻近算子的内积。同时，我们从惩罚次梯度的性质出发，刻画了邻近估计量的Oracle特性。我们利用该方法系统性地覆盖了规则与不规则设计下的线性回归设定。针对这些设定，我们构建了新的$\sqrt{n}-$相合、渐近正态的Ridgeless型邻近估计量，这些估计量具备Oracle性质，并在实际相关的蒙特卡洛模拟中表现出令人满意的性能。