Influence functions for penalized M-estimators

We study the local robustness properties of general nondifferentiable penalized M-estimators via the influence function. More precisely, we propose a framework that allows us to define rigorously the influence function as the limiting influence function of a sequence of approximating estimators. We show that it can be used to characterize the robustness properties of a wide range of sparse estimators and we derive its form for general penalized M-estimators including lasso and adaptive lasso type estimators. We prove that our influence function is equivalent to a derivative in the sense of distribution theory.