Note on Qualitative Robustness of Multivariate Sample Mean and Median

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 -asymptotic normality of estimators in , and examine such robustness of two standard location estimators in . 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 -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 -sample median could not be “qualitatively robust” with respect to the same distance between the distributions. 1. Introduction The following Hampel’s definition (originally given for the one-dimensional case) of qualitative robustness [1, 2] deals with -balls in the space of distributions rather than with standard “contamination neighborhoods” (see for the latter, e.g., [3, 4]). The sequence , , of estimators is qualitatively robust at the distribution if for every there exists such that entails Here, and throughout, denotes the Prokhorov metric, and ; , are i.i.d. random vectors distributed, respectively, as and . For a metric on the space of distributions and random vectors we will write (as in (1)) having in mind the -distance between the distributions of and . By all means, the use of the Prokhorov metric is only an option. For instance, in [5] other probability metrics in the definition of qualitative robustness were used. (See also [6–9] for using different probability metrics or pseudo-metrics related to the estimation of robustness). As noted in [1, 2] in , sample means are not qualitatively robust at any , while sample medians are qualitatively robust at any having a unique median. (See also [10] for lack of qualitative robustness of sample means in certain Banach spaces). Moreover, in [11] it was shown that for symmetric distributions the median is, in certain sense, the “most robust” estimator of a center location when using the pseudo-metric corresponding to neighborhoods of contamination type. (See more with this respect in [8]). At the same time, it is known from the literature that under different circumstances, in particular, using distinct probability metrics (as, e.g., in (1) or in other