L-estimation for linear models

Linear combinations of order statistics, or L-estimators, have played an extremely important role in the development of robust methods for the one-sample problem. We suggest analogs of L-estimators for the parameters of the linear model based on the p-dimensional “regression quantiles” proposed by Koenker and Bassett (1978). A uniform, Bahadur-type asymptotic representation of regression quantiles is established, and this yields a general asymptotic theory of L-estimators for the linear model. A leading example of the proposed estimators is an analog of the trimmed mean (TRQ), which is asymptotically equivalent to the trimmed least squares estimator studied by Ruppert and Carroll (1980), but appears to be somewhat less sensitive to influential design observations. This estimator is also asymptotically equivalent to the well-known Huber M-estimator, but offers the significant advantage that it is scale invariant. We illustrate the methods by reconsidering a mid-18th century linear model analyzed by Boscovich. It is apparent that the proposed methods yield estimators that are weighted averages of coefficient vectors determined by certain p-element subsets of the n sample observations. The subset of p-element subsets that generate solutions to the regression quantile optimization problem play the role of order statistics for the linear model and may be useful in other applications. We also investigate two proposals for estimating the covariance matrix for the trimmed regression quantile estimator. One approach employs residuals from the TRQ fit to estimate a winsorized variance, the other employs the empirical quantile function suggested in Bassett and Koenker (1982). Using the Monte Carlo methods of Gross (1977) we find that either approach yields test statistics with critical values close to those of the conventional t test and good expected confidence interval lengths.