Lower Bounds for the Asymptotic Relative Efficiency of Huber Regression

Huber regression serves as a prominent robust alternative to ordinary least squares (OLS), particularly in the presence of heavy-tailed error distributions. While the asymptotic relative efficiency (ARE) of Huber regression is well documented for the standard normal distribution, its worst-case efficiency across the class of all continuous and symmetric error distributions remains an important theoretical question. In this paper, we establish positive lower bounds for the ARE of Huber regression relative to OLS. By strategically selecting the robustification parameter based on the moments or quantiles of the error distribution, we first prove that the ARE is uniformly bounded away from zero across all continuous and symmetric error distributions. This result guarantees a baseline level of efficiency for Huber regression, sharing a similar theoretical spirit with the celebrated lower bound of the Wilcoxon rank estimator. Utilizing the empirical process theory, we further establish that the relative efficiency of Huber regression remains unchanged if the theoretical tuning parameter is replaced by an estimator with a suitable convergence rate. Simulation studies are conducted to examine the performance of Huber regression under the proposed tuning strategies.