Note on qualitative robustness of multivariate sample mean and median (Q1952472)

Summary: 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 \(\sqrt{n}\)-asymptotic normality of estimators in \(\mathbb R^k\), and examine such robustness of two standard location estimators in \(\mathbb R^k\). 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 \(L_1\)-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 \(L_1\)-sample median could not be ``qualitatively robust'' with respect to the same distance between the distributions.