Multivariate trimmed means based on the Tukey depth |
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Authors: | Jean-Claude Massé |
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Institution: | Département de Mathématiques et de Statistique, Université Laval, Sainte-Foy, QC, Canada G1K 7P4 |
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Abstract: | In univariate statistics, the trimmed mean has long been regarded as a robust and efficient alternative to the sample mean. A multivariate analogue calls for a notion of trimmed region around the center of the sample. Using Tukey's depth to achieve this goal, this paper investigates two types of multivariate trimmed means obtained by averaging over the trimmed region in two different ways. For both trimmed means, conditions ensuring asymptotic normality are obtained; in this respect, one of the main features of the paper is the systematic use of Hadamard derivatives and empirical processes methods to derive the central limit theorems. Asymptotic efficiency relative to the sample mean as well as breakdown point are also studied. The results provide convincing evidence that these location estimators have nice asymptotic behavior and possess highly desirable finite-sample robustness properties; furthermore, relative to the sample mean, both of them can in some situations be highly efficient for dimensions between 2 and 10. |
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Keywords: | Asymptotic relative efficiency Breakdown point Functional delta method Hadamard derivative Multivariate trimmed mean Robustness Trimmed region Tukey depth |
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