Formulating robust regression estimation as an optimum allocation problem

Abstract

The Mallows-type estimator, one of the most reasonable bounded influence estimators, often downweights leverage points regardless of the magnitude of the corresponding residual, and this could imply a loss of efficiency. In this article, we consider whether the efficiency of this bounded influence estimator could be improved by regarding both the robust x -distance and the residual size. We develop a new robust procedure based on the ideas of the Mallows-type estimator and the general robust recipe, where data been cleaned by pulling outliers towards their fitted values. Our basic idea is to formulate the robust estimation as an allocation problem, where the objective function is a Huber-type "loss" function, but the pulling resource is restricted. Using a mathematical programming technique, the pulling resource is optimally allocated to influential points <$>({x}_i, y_i)<$> with respect to residual size and given weights, <$>w({x}_i)<$>. Three previously published approaches are compared to our proposal via simulated experiments. In the case of contaminated data by regression outliers and "good" leverage points, the proposed robust estimator is a reasonable bounded influence estimator concerning both efficiency and norm of bias. In addition, the proposed approach offers the potential to establish constraints for the regression parameters and also may potentially provide insight regarding outlier detection.