Robustness of weighted L p–depth and L p–median |
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Authors: | Yijun Zuo |
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Institution: | (1) Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA |
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Abstract: | Summary:
L
p
–norm weighted depth functions are introduced and the local and global robustness of these weighted L
p
–depth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted L
p
–depth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted L
p
–depth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted L
p
–medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted L
p
depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted L
p
depth and medians, respectively.*The author thanks the referees for their very insightful and constructive comments
and suggestions which led to corrections and substantial improvements. Supported in
part by NSF Grants DMS-0071976 and DMS-0134628. |
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Keywords: | Breakdown point depth function efficiency equivariance influence function L p – norm median robustness |
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