Consistency and asymptotic distribution of the Theil–Sen estimator

Abstract

In this paper, we obtain the strong consistency and asymptotic distribution of the Theil–Sen estimator in simple linear regression models with arbitrary error distributions. We show that the Theil–Sen estimator is super-efficient when the error distribution is discontinuous and that its asymptotic distribution may or may not be normal when the error distribution is continuous. We give an example in which the Theil–Sen estimator is not asymptotically normal. A small simulation study is conducted to confirm the super-efficiency and the non-normality of the asymptotic distribution.