Reweighted LS Estimators Converge at the same Rate as the Initial Estimator

The problem of combining high efficiency with high breakdown properties for regression estimators has piqued the interest of statisticians for some time. One proposal specifically suggested by Rousseeuw and Leroy is to use the least median of squares estimator, omit observations whose residuals are larger than some constant cut-off value and apply least squares to the remaining observations. Although this proposal does retain high breakdown point, it actually converges no faster than the initial estimator. In fact, the reweighted least squares estimator is asymptotically a constant times the initial estimator if the initial estimator converges at a rate strictly slower than $n^{-1/2}$.