Regression-Free and Robust Estimation of Scale for Bivariate Data

In this paper we present robust estimators for the dispersion of the errors in simple linear regression. Existing scale estimators are based on the residuals from an estimator of the regression itself. Instead, we propose scale estimators that do not depend on any previous estimate of the regression parameters. For this purpose we consider triangles formed by data points, and deene their vertical height. Taking the repeated median of all such heights leads to a 50% breakdown point estimator. A second estimator is obtained from the 0:278-quantile of all triangle heights, and results in a breakdown point of 34:7%. When we restrict ourselves to the heights of adjacent triangles and take their 0:4-quantile, we obtain a much faster estimator with a 20% breakdown point. Simulations are carried out to study the computation time and statistical performance of these estimators.