Note on Qualitative Robustness of Multivariate Sample Mean and Median

It is known that the robustness properties of estimators dependon the choice of a metric in the space of distributions. We introducea version of Hampel's qualitative robustness that takes into account the n-asymptotic normality of estimators in Rk, and examine suchrobustness of two standard location estimators in ℝk. For this purpose, we use certain combination of the Kantorovich and Zolotarevmetrics rather than the usual Prokhorov type metric. This choice ofthe 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 probabilitymetrics we show the qualitative robustness of the sample multivariatemean and prove the inequality which provides a quantitative measureof robustness. On the other hand, we show that L1-sample median could not be “qualitatively robust” with respect to the same distancebetween the distributions