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Finite sample breakdown point of Tukey’s halfspace median

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Abstract

Tukey’s halfspace median (HM), servicing as the multivariate counterpart of the univariate median, has been introduced and extensively studied in the literature. It is supposed and expected to preserve robustness property (the most outstanding property) of the univariate median. One of prevalent quantitative assessments of robustness is finite sample breakdown point (FSBP). Indeed, the FSBP of many multivariate medians have been identified, except for the most prevailing one—the Tukey’s halfspace median. This paper presents a precise result on FSBP for Tukey’s halfspace median. The result here depicts the complete prospect of the global robustness of HM in the finite sample practical scenario, revealing the dimension effect on the breakdown point robustness and complimenting the existing asymptotic breakdown point result.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11601197, 11461029, 71463020, 61263014 and 61563018), National Science Foundation of Jiangxi Province (Grant Nos. 20161BAB201024, 20142BAB211014, 20143ACB21012 and 20151BAB211016), the Key Science Fund Project of Jiangxi Provincial Education Department (Grant Nos. GJJ150439, KJLD13033 and KJLD14034), the National Science Fund for Distinguished Young Scholars in China (Grant No. 10725106), the National Natural Science Foundation of China (Grant Nos. General program 11171331 and Key program 11331011), a grant from the Key Lab of Random Complex Structure and Data Science, Chinese Academy of Sciences and Natural Science Foundation of Shenzhen University. The authors thank the two anonymous referees and the AE for their careful reading and insightful comments, which led to many improvements of this paper.

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Authors and Affiliations

  1. School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    XiaoHui Liu

  2. Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    XiaoHui Liu

  3. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48823, USA

    YiJun Zuo

  4. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    QiHua Wang

  5. Institute of Statistical Science, Shenzhen University, Shenzhen, 518006, China

    QiHua Wang

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  1. XiaoHui Liu
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  2. YiJun Zuo
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  3. QiHua Wang
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Correspondence to XiaoHui Liu.

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Liu, X., Zuo, Y. & Wang, Q. Finite sample breakdown point of Tukey’s halfspace median. Sci. China Math. 60, 861–874 (2017). https://doi.org/10.1007/s11425-016-0285-1

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