The Influence Function of Penalized Regression Estimators

TL;DR

This paper analyzes the robustness of penalized regression estimators by computing their influence functions, asymptotic variance, and mean squared error, highlighting that only M-estimators with bounded derivatives are robust.

Contribution

It provides the first detailed computation of influence functions for penalized M-estimators and sparse LTS, revealing robustness conditions and limitations of popular methods like lasso.

Findings

Lasso has an unbounded influence function, indicating non-robustness.

Only M-estimators with bounded derivative loss functions are robust.

The paper derives asymptotic variance and MSE for these estimators.

Abstract

To perform regression analysis in high dimensions, lasso or ridge estimation are a common choice. However, it has been shown that these methods are not robust to outliers. Therefore, alternatives as penalized M-estimation or the sparse least trimmed squares (LTS) estimator have been proposed. The robustness of these regression methods can be measured with the influence function. It quantifies the effect of infinitesimal perturbations in the data. Furthermore it can be used to compute the asymptotic variance and the mean squared error. In this paper we compute the influence function, the asymptotic variance and the mean squared error for penalized M-estimators and the sparse LTS estimator. The asymptotic biasedness of the estimators make the calculations nonstandard. We show that only M-estimators with a loss function with a bounded derivative are robust against regression outliers. In particular, the lasso has an unbounded influence function.