The Theil–Sen estimator in a measurement error perspective

Abstract

In a simple measurement error regression model, the classical least squares estimator of the slope parameter consistently estimates a discounted slope, though sans normality, some other properties may not hold. It is shown that for a broader class of error distributions, the Theil–Sen estimator, albeit nonlinear, is a median-unbiased, consistent and robust estimator of the same discounted parameter. For a general class of nonlinear (including R–, M– and L– estimators), study of asymptotic properties is greatly facilitated by using some uniform asymptotic linearity results, which are, in turn, based on contiguity of probability measures. This contiguity is established in a measurement error model under broader distributional assumptions. Some asymptotic properties of the Theil–Sen estimator are studied under slightly different regularity conditions in a direct way bypassing the contiguity approach.