Towards a General Large Sample Theory for Regularized Estimators

TL;DR

This paper introduces a comprehensive framework for analyzing regularized estimators, establishing their consistency and asymptotic properties, and providing data-driven tuning methods applicable across diverse estimation problems.

Contribution

It offers a unified theoretical approach for regularized estimators, including consistency, asymptotic linearity, and practical tuning strategies, broadening their applicability.

Findings

Established general conditions for consistency of regularized estimators.

Derived a generalized asymptotic linearity property.

Proposed data-driven tuning parameter selection methods.

Abstract

We present a general framework for studying regularized estimators; such estimators are pervasive in estimation problems wherein "plug-in" type estimators are either ill-defined or ill-behaved. Within this framework, we derive, under primitive conditions, consistency and a generalization of the asymptotic linearity property. We also provide data-driven methods for choosing tuning parameters that, under some conditions, achieve the aforementioned properties. We illustrate the scope of our approach by presenting a wide range of applications.