Influence function and maximum bias of projection depth based estimators

Abstract

Location estimators induced from depth functions increasingly have been pursued and studied in the literature. Among them are those induced from projection depth functions. These projection depth based estimators have favorable properties among their competitors. In particular, they possess the best possible finite sample breakdown point robustness. However, robustness of estimators cannot be revealed by the finite sample breakdown point alone. The influence function, gross error sensitivity, maximum bias and contamination sensitivity are also important aspects of robustness. In this article, we study these other robustness aspects of two types of projection depth based estimators: projection medians and projection depth weighted means. The latter includes the Stahel-Donoho estimator as a special case. Exact maximum bias, the influence function, and contamination and gross error sensitivity are derived and studied for both types of estimators. Sharp upper bounds for the maximum bias and the influence functions are established. Comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors.