Effect of leverage on the finite sample efficiencies of high breakdown estimators

The presence of leverage points in regression can have a dramatic impact on the finite sample efficiencies (versus least squares under Gaussian errors) of high breakdown estimators. In general, as the x-value of a leverage point becomes more extreme, the finite sample efficiencies decrease and larger sample sizes are required for asymptotic results to apply. However, the least median of squares (LMS) and least trimmed squares (LTS) estimators exhibit higher finite sample than asymptotic efficiencies. On the other hand, the finite sample efficiency of the Schweppe-type one-step generalized M estimator (S1S), starting from LTS, which minimizes the sum of the h smallest squared residuals, converges from below to its asymptotic value as the sample size increases. We suggest that the finite sample variance itself be replaced by a trimmed sample variance which accounts for the long tail behavior encountered in finite samples. An improvement on the S1S estimator, which consists of never downweighting any point with one of the h smallest squared LTS residuals, performs quite well with respect to either measure of efficiency, while retaining a high breakdown point.