Tail Behavior of Regression Estimators and their Breakdown Points

A measure of finite sample estimator performance based on the probability of large deviations is introduced. The tail performance of the least squares estimator is studied in detail, and the authors find that it achieves good tail performance under strictly Gaussian conditions, but performance is extremely poor in the case of heavy-tailed error distributions. Turning to the tail behavior of various robust estimators, they focus on tail performance under heavy (algebraic) tail errors. Perhaps most significantly, it is shown that the authors' finite-sample measure of tail performance is, for heavy tailed error distributions, essentially the same as the finite sample concept of breakdown point introduced by D. L. Donoho and P. J. Huber (1983). Coauthors are Jana Jureckova, Roger Koenker, and Stephen Portnoy.