The Fast-τ Estimator for Regression

Yohai and Zamar's τ-estimators of regression have excellent statistical properties but are nevertheless rarely used in practice because of a lack of available software and the general impression that τ-estimators are difficult to approximate. We will show, however, that the computational difficulties of approximating τ-estimators are similar in nature to those of the more popular S-estimators. The main goal of this article is to compare an approximating algorithm for τ-estimators based on random resampling with some alternative heuristic search algorithms. We show that the former is not only simpler, but that when enhanced by local improvement steps it generally outperforms the consider edheuristic search algorithms, even when the seheuristic algorithms also incorporate local improvement steps. Additionally, we show that the random resampling algorithm for approximating τ-estimators has favorable statistical properties compared to the analogous and widely used algorithms for S- and least trimmed squares estimators.