On Robust Estimation of Error Variance in (Highly) Robust Regression

Abstract

The linear regression model requires robust estimation of parameters, if the measured data are contaminated by outlying measurements (outliers). While a number of robust estimators (i.e. resistant to outliers) have been proposed, this paper is focused on estimating the variance of the random regression errors. We particularly focus on the least weighted squares estimator, for which we review its properties and propose new weighting schemes together with corresponding estimates for the variance of disturbances. An illustrative example revealing the idea of the estimator to down-weight individual measurements is presented. Further, two numerical simulations presented here allow to compare various estimators. They verify the theoretical results for the least weighted squares to be meaningful. MM-estimators turn out to yield the best results in the simulations in terms of both accuracy and precision. The least weighted squares (with suitable weights) remain only slightly behind in terms of the mean square error and are able to outperform the much more popular least trimmed squares estimator, especially for smaller sample sizes.