Efficient Robust Estimation of Regression Models

Abstract

This paper introduces a new class of robust regression estimators. The proposed twostep least weighted squares (2S-LWS) estimator employs data-adaptive weights determined from the empirical distribution, quantile, or density functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However contrary to existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions; most importantly, the initial estimator does not need to be square root of n consistent. Moreover, we prove that 2S-LWS is asymptotically normal under beta-mixing conditions and asymptotically efficient if errors are normally distributed. A simulation study documents these theoretical properties in finite samples; in particular, the relative efficiency of 2S-LWS can reach 85-90% in samples of several tens of observations under various distributional models.