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- W2046729048 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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:mrow><mml:msqrt><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msqrt></mml:mrow></mml:math>-asymptotic normality of estimators in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math>, and examine such robustness of two standard location estimators in<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:math>. 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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math>-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<mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M5><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math>-sample median could not be “qualitatively robust” with respect to the same distance between the distributions." @default.
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- W2046729048 date "2013-01-01" @default.
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- W2046729048 title "Note on Qualitative Robustness of Multivariate Sample Mean and Median" @default.
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- W2046729048 doi "https://doi.org/10.1155/2013/208950" @default.
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