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Journal of Probability and Statistics
Volume 2013, Article ID 208950, 8 pages
http://dx.doi.org/10.1155/2013/208950
Research Article

Note on Qualitative Robustness of Multivariate Sample Mean and Median

Department of Mathematics, Autonomous Metropolitan University, Iztapalapa, San Rafael Atlixco 186, Col. Vicentina, C.P. 09340, Mexico City, DF, Mexico

Received 4 October 2012; Revised 11 December 2012; Accepted 12 December 2012

Academic Editor: Shein-chung Chow

Copyright © 2013 Evgueni Gordienko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -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.