A Note on the Theil-Sen Regression Estimator When the Regressor Is Random and the Error Term Is Heteroscedastic

In simple regression, two serious problems with the ordinary least squares (OLS) estimator are that its efficiency can be relatively poor when the error term is normal but heteroscedastic, and the usual confidence interval for the slope can have highly unsatisfactory probability coverage. When the error term is nonnormal, these problems become exacerbated. Two other concerns are that the OLS estimator has an unbounded influence function and a breakdown point of zero. Wilcox (1996) compared several estimators when there is heteroscedasticity and found two that have relatively good efficiency and simultaneously provide protection against outliers: an M-estimator with Schweppe weights and an estimator proposed by Cohen, Dalal and Tukey (1993). However, the M-estimator can handle only one outlier in the X-domain or among the Y values, and among the methods considered by Wilcox for computing confidence intervals for the slope, none performed well when working with the Cohen-Dalal-Tukey estimator. This note points out that the small-sample efficiency of theTheil-Sen estimator competes well with the estimators considered by Wilcox, and a method for computing a confidence interval was found that performs well in simulations. The Theil-Sen estimator has a reasonably high breakdown point, a bounded influence function, and in some cases its small-sample efficiency offers a substantial advantage over all of the estimators compared in Wilcox (1996).