Unbiasedness of the Theil–Sen estimator

Abstract

We consider the simple linear regression model. The Theil–Sen estimator is a point estimator of the slope parameter in the model and has many nice properties, most of which are established by Sen. Thus, it is introduced in several classical textbooks on non-parametric statistics. Sen also gave a proof that the Theil–Sen estimator is unbiased under the assumption that the error distribution is continuous. The statement is incorrect. We construct several counterexamples. Furthermore, we show that the continuity assumption on the error distribution is not important to unbiasedness. In particular, if the sample size n = 2 or 3, then the Theil–Sen estimator is unbiased. Moreover, if either the error distribution or the covariates have certain symmetry, then the Theil–Sen estimator is also unbiased.