Robust, location-free scale estimators for the linear regression and k-sample models

Abstract

In the last few years, estimators of the scale of a univariate distribution have been developed that are location-free in the sense that they do not depend on an estimate of the center of the underlying distribution. These proposed location-free estimators have generally been quite robust in terms of having a high breakdown point and can achieve a surprisingly high Gaussian efficiency. This idea has also been extended to the simple linear regression model, where typical estimators of the dispersion of the errors depend on an estimator of the regression line. The few estimators that have been developed that do not depend on a line estimator, called regression-free scale estimators, do achieve a high breakdown point but are useful mainly for data sets that have no replication at any regressor value. We propose new regression-free scale estimators that achieve a high breakdown point, can be quite efficient, and are useful when the data contain replication. Also, we propose a robust estimator of the common scale parameter in the k-sample model that reduces to an existing location-free estimator in the case of univariate data. We derive the breakdown point of this estimator as well as its maximum bias curve. Simulation results show that it can be quite efficient with Gaussian data.