A class of robust and fully efficient regression estimators

TLDR

This paper introduces a new class of robust estimators for the linear regression model. They are weighted least squares estimators, with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator. The estimators achieve asymptotic breakdown points at least as high as the initial estimator, have finite‑sample breakdown points at most 1/n lower, exhibit negligible bias increase for least‑median‑of‑squares initial estimators under normal errors, attain full asymptotic efficiency under normal errors and general design, and outperform the initial estimators in finite‑sample Monte Carlo studies.

Abstract

This paper introduces a new class of robust estimators for the linear regression model. They are weighted least squares estimators, with weights adaptively computed using the empirical distribution of the residuals of an initial robust estimator. It is shown that under certain general conditions the asymptotic breakdown points of the proposed estimators are not less than that of the initial estimator, and the finite sample breakdown point can be at most $1/n$ less. For the special case of the least median of squares as initial estimator, hard rejection weights and normal errors and carriers, the maximum bias function of the proposed estimators for point-mass contaminations is numerically computed, with the result that there is almost no worsening of bias. Moreover–and this is the original contribution of this paper–if the errors are normally distributed and under fairly general conditions on the design the proposed estimators have full asymptotic efficiency. A Monte Carlo study shows that they have better behavior than the initial estimators for finite sample sizes.