Skip to main content
SpringerLink
Log in

Large sample and robust properties of \( \tilde L^2 \)-median

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

As a competitive depth, \( \tilde L^2 \)-depth is modified from L 2-depth. Its induced median is called \( \tilde L^2 \)-median. Basic properties of the median and its sample version are provided. Especially, the strong consistency of sample median is gained under weaker condition. Robustness of the median and its sample version is discussed. Besides ease of computation, it is shown that \( \tilde L^2 \)-median has both good large-sample and robust properties. Simulation studies are also given to compare the breakdown point of \( \tilde L^2 \)-median with that of other depth-induced medians.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Y. Liu, J. M. Parelius, and K. Singh, Multivariate analysis by data depth: Descriptive statistics, graphics and inference (with discussion), Ann. Statist., 1999, 27(3): 783–858.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Y. Liu and K. Singh, A quality index based on data depth and multivariate rank tests, J. Amer. Statist. Assoc., 1993, 88: 252–260.

    Article  MATH  MathSciNet  Google Scholar 

  3. P. J. Rousseeuw and M. Hubert, Regression depth (with discussion), J. Amer. Statist. Assoc., 1999, 94: 388–433.

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Jörnsten, et al., A robust clustering method and visualization tool based on data depth, statistical data analysis based on the L 1 norm and related methods, Statistics for Industry and Technology, 2002, 27(3): 783–858.

    Google Scholar 

  5. R. Jörnsten, Clustering and classification based on the L 1 data depth, JMA, 2004, 90: 67–89.

    MATH  Google Scholar 

  6. Y. Zuo and H. Cui, Statistical depth functions and some applications, Advances in Mathematics, 2004, 33(1): 1–25.

    Article  MathSciNet  Google Scholar 

  7. G. Small, A survey of multidimensional medians, International Statistical Review, 1990, 58(3): 263–277.

    Article  Google Scholar 

  8. Y. Zuo and R. Serfling, On the performance of some nonparametric location measures relative to a general notion of multivariate symmetry, Journal of Statistical Planning and Inference, 2000, 84: 55–79.

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. Zuo, General notions of statistical depth function, The Annals of Statistics, 2000, 28(2): 461–482.

    Article  MATH  MathSciNet  Google Scholar 

  10. B. Brown, Statistical uses of the spatial median, JRSS B, 1983, 45(1): 25–30.

    MATH  Google Scholar 

  11. C. R. Rao, Methodology based on the L 1 norm in statistical inference, Sankhyā Ser. A, 1988, 50: 289–313.

    MATH  Google Scholar 

  12. T. Isogai, Some extension of haldane’s multivariate median and its application, Ann. Inst. Statist. Math., 1985, 37(A): 289–301.

    Article  MATH  MathSciNet  Google Scholar 

  13. Y. Vardi and C. Zhang, The Multivariate L 1-median and associated data depth, Proc. Nat. Acad. Sci. USA, 2000, 97(4): 1423–1426.

    Article  MATH  MathSciNet  Google Scholar 

  14. B. Chakraborty, P. Chaudhuri, and H. Oja, Operating transformation retransformation on spatial median and angle test, Statistical Sinica, 1998, 8: 767–784.

    MATH  MathSciNet  Google Scholar 

  15. B. Chakraborty and P. Chaudhuri, A note on the robustness of multivariate medians, Statistics & Probability Letters, 1999, 45: 269–276.

    Article  MATH  MathSciNet  Google Scholar 

  16. Y. Zuo, Robustness of weighted L p-depth and L p-median, Allgemeines Statistiches Archiv, 2004, 88: 1–20.

    Article  MathSciNet  Google Scholar 

  17. T. F. Xie and G. Y. Li, Basic properties of \( \tilde L^2 \) sample depth, Acta Mathematicae Applicatae Sinica, 2006, 29(5): 893–900.

    MathSciNet  Google Scholar 

  18. R. Y. Liu, On a notion of data depth based on random simplices, Annals of Statistics, 1990, 18: 405–414.

    Article  MATH  MathSciNet  Google Scholar 

  19. J. B. S. Haldane, Note on the median of a multivariate distribution, Biometrika, 1948, 35: 414–415.

    MATH  MathSciNet  Google Scholar 

  20. P. Chaudhuri, Multivariate location estimation using extension of R-estimates through U-statistics type approach, Annals of Statistics, 1992, 20: 897–916.

    Article  MATH  MathSciNet  Google Scholar 

  21. F. R. Hampel, E. Z. Ronchetti, P. J. Rousseeuw, and W. A. Stahel, Robust Statistics: The Approach Based on Influence Functions, Wiley, New York, 1986.

    MATH  Google Scholar 

  22. K. T. Fang and Y. Zhang, Generalized Multivariate Analysis, Springer and Science Press, 1990.

  23. J. Zhang and G. Li, Breakdown properties of M-estimators of location, Ann. Statist., 1998, 26: 1170–1189.

    Article  MATH  MathSciNet  Google Scholar 

  24. D. L. Donoho, Breakdown properties of multivariate location estimators, Ph.D. Thesis, Harvard University, Boston, 1982.

    Google Scholar 

  25. P. Lopuhaä and P. Rousseeuw, Breakdown points of Affine equivariant estimators of multivariate location and covariance matrices, Ann. Statist., 1991, 19(1): 229–248.

    Article  MATH  MathSciNet  Google Scholar 

  26. P. Lopuhaä, Asymptotic expansion of S-estimators of location and covariance, Statist. Neerlandica, 1997, 51(2): 220–237.

    Article  MATH  MathSciNet  Google Scholar 

  27. B. Kemperman, The median of a finite measure on a Banach space, Statistical Data Analysis Based on the L 1 -Norm and Related Methods, Y. Dodge, 1987: 217–230.

  28. G. Li and J. Zhang, Sphering and its properties, Sankhyā: The Indian Journal of Statistics, 1998, 60A(1): 119–133.

    Google Scholar 

  29. T. F. Xie, Two statistical data depth and their medians, Ph.D Thesis, Academy of Mathematics and Systems Science, Chinese Academy of Science, 2005, Beijing.

    Google Scholar 

  30. R. Serfling, Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardization, Journal of Nonparametric Statistics, 2010, 22: 915–936.

    Article  Google Scholar 

  31. P. J. Rousseeuw and K. V. Driessen, A fast algorithm for the minimum covariance determinant estimator, Technometrics, 1999, 41: 212–223.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China

    Tianfa Xie

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

    Guoying Li

Authors
  1. Tianfa Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guoying Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tianfa Xie.

Additional information

This research is supported by the National Natural Science Foundation of China under Grant No. 10971007, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20091103120012, and the research fund of BJUT under Grant No. X0006013200904.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Xie, T., Li, G. Large sample and robust properties of \( \tilde L^2 \)-median. J Syst Sci Complex 23, 1133–1142 (2010). https://doi.org/10.1007/s11424-010-8423-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-010-8423-z

Key words

Search

Navigation