Note on Qualitative Robustness of Multivariate Sample Mean and Median

Abstract

It is known that the robustness properties of estimators depend on the choice of a metric in the space of distributions. We introduce a version of Hampel's qualitative robustness that takes into account the n-asymptotic normality of estimators in Rk, and examine such robustness of two standard location estimators in ℝk. For this purpose, we use certain combination of the Kantorovich and Zolotarev metrics rather than the usual Prokhorov type metric. This choice of the metric is explained by an intention to expose a (theoretical) situation where the robustness properties of sample mean and L1-sample median are in reverse to the usual ones. Using the mentioned probability metrics we show the qualitative robustness of the sample multivariate mean and prove the inequality which provides a quantitative measure of robustness. On the other hand, we show that L1-sample median could not be “qualitatively robust” with respect to the same distance between the distributions.