Trimmed Least Squares

We consider two methods of defining a regression analogue to a trimmed mean. The first was suggested by Koenker and Bassett and uses their concept of regression quantiles. Its asymptotic behavior is completely analogous to that of a trimmed mean. The second method uses residuals from a preliminary estimator. Its asymptotic behavior depends heavily on the preliminary estimate; it behaves, in general, quite differently than the estimator proposed by Koenker and Bassett, and it can be rather inefficient at the normal model even if the percent trimming is small. However, if the preliminary estimator is the average of the two regression quantiles used with Koenker and Bassett's estimator, then the first and second methods are asymptotically equivalent for sYmmetric error distributions. •