L-Estimation for Linear Models

Abstract

Abstract 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 b...