Highly Robust and Highly Finite Sample Efficient Estimators for the Linear Model

Abstract

In this paper, we propose a new family of robust regression estimators, which we call bounded residual scale estimators (BRS-estimators) which are simultaneously highly robust and highly efficient for small samples with normally distributed errors. To define these estimators it is required to have a robust M-scale and a family of robust MM-estimators. We start by choosing in this family a highly robust initial estimator but not necessarily highly efficient. Loosely speaking, the BRS-estimator is defined as the estimator in the MM family which is closest to the LSE among those with a robust M-scale sufficiently close to the one of the initial estimators. The efficiency of the BRS is derived from the fact that when there are not outliers in the sample and the errors are normally distributed, the scale of the LSE is similar to the one of the initial estimator. The robustness of the BRS-estimator comes from the fact that its robust scale is close to the one of the initial highly robust estimator. The results of a Monte Carlo study show that the proposed estimator has a high finite-sample efficiency, and is highly resistant to outlier contamination.