Robust Weighted LAD Regression

Abstract

The least squares linear regression estimator is well-known to be highly sensitive to unusual observations in the data, and as a result many more robust estimators have been proposed as alternatives. One of the earliest proposals was least-sum of absolute deviations (LAD) regression, where the regression coefficients are estimated through minimization of the sum of the absolute values of the residuals. LAD regression has been largely ignored as a robust alternative to least squares, since it can be strongly affected by a single observation (that is, it has a breakdown point of 1/n, where n is the sample size). In this paper we show that judicious choice of weights can result in a weighted LAD estimator with much higher breakdown point. We discuss the properties of the weighted LAD estimator, and show via simulation that its performance is competitive with that of high breakdown regression estimators, particularly in the presence of outliers located at leverage points. We also apply the estimator to several real data sets.