Robust Regression Estimators When There are Tied Values

Abstract

It is well known that when using the ordinary least squares regression estimator, outliers among the dependent variable can result in relatively poor power. Many robust regression estimators have been derived that address this problem, but the bulk of the results assume that the dependent variable is continuous. It is demonstrated that when there are tied values, several robust regression estimators can perform poorly in terms of controlling the Type I error probability, even with a large sample size. The presence of tied values does not necessarily mean that they perform poorly, but there is the issue of whether there is a robust estimator that performs reasonably well in situations where other estimators do not. The main result is that a modification of the Theil–Sen estimator achieves this goal. Results on the small-sample efficiency of the modified Theil–Sen estimator are reported as well. Data from the Well Elderly 2 Study, which motivated this study, are used to illustrate that the modified Theil–Sen estimator can make a practical difference.