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Multivariate Trimmed Means Based on Data Depth

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Statistical Data Analysis Based on the L1-Norm and Related Methods

Part of the book series: Statistics for Industry and Technology ((SIT))

Abstract

Multivariate analogues of the univariate median have been successfully introduced based on data depth. Like their univariate counterpart, these multivariate medians in general are quite robust but not very efficient. Univariate trimmed means, which can keep a desirable balance between robustness and efficiency, are known to be the alternatives to the univariate median. Multivariate analogues of univariate trimmed means can also be introduced based on data depth. In this article, we study multivariate depth trimmed means with a main focus on their limiting distribution, robustness, and efficiency.

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References

  1. Arcones, M. A., Chen, Z., and Giné, E. (1994). Estimators related to U-processes with applications to multivariate medians: asymptotic normality. Ann. Statist. 22, 1460–1477.

    Article  MATH  MathSciNet  Google Scholar 

  2. Bickel, P. J. (1965). On some robust estimates of location. Ann. Math. Statist. 36, 847–858.

    Article  MATH  MathSciNet  Google Scholar 

  3. Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. qualifying paper, Dept. Statistics, Harvard University.

    Google Scholar 

  4. Donoho, D. L., and Gasko, M. (1992). Breakdown properties of multivariate location parameters and dispersion matrices. Ann. Statist. 20 1803–1827.

    Article  MATH  MathSciNet  Google Scholar 

  5. Donoho, D. L., and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift foe Erich L. Lehmann (P. J. Bickel, K. A. Doksum and J. L. Hodges, Jr., eds.) 157–184. Wadsworth, Belmont, CA.

    Google Scholar 

  6. Dümbgen, L. (1992). Limit theorem for the simplicial depth. Statist. Probab. Letters. 14 119–128.

    Article  MATH  Google Scholar 

  7. Einmahl, J. H. J., and Mason, D. M. (1992). Generalized quantile processes. Ann. Statist. 20 1062–1078.

    Article  MATH  MathSciNet  Google Scholar 

  8. Kim, S. J. (1992). The metrically trimmed mean as a robust estimator of location. Ann. Statist 20 1534–1547.

    Article  MATH  MathSciNet  Google Scholar 

  9. Kim, J. (2000). Rate of convergence of depth contours: with application to a multivariate metrically trimmed mean. Statist. Probab. Letters 49 393–400.

    Article  MATH  Google Scholar 

  10. Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist 18 405–414.

    Article  MATH  MathSciNet  Google Scholar 

  11. Liu, R. Y. (1992). Data Depth and Multivariate Rank Tests. In L 1-Statistical Analysis and Related Methods (Y. Dodge, ed.) 279-294.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  13. Massé, J. C. (1999). Asymptotics for the Tukey depth. Preprint.

    Google Scholar 

  14. Mosteller, C. F. and Tukey, J. W. (1977). Data Analysis and Regression. Addison-Wesley, Reading, Mass.

    Google Scholar 

  15. Pollard, D. (1984). Convergence of Stochastic Processes. Springer-Verlag.

    Google Scholar 

  16. Rousseeuw, P. J., and Ruts, I. (1996). AS 307: Bivariate location depth. Applied Statist. 45 516–526.

    Article  MATH  Google Scholar 

  17. Rousseeuw, P. J., and Ruts, I. (1998). Constructing the bivariate Tukey median. Statist. Sinica 8 828–839.

    MathSciNet  Google Scholar 

  18. Serfling, R. (1980). Approximation Theorems of Mathematical Statistics. John Wiley & Sons.

    Google Scholar 

  19. Serfling, R. (2002). Generalized quantile processes based on multivariate depth functions, with applications in nonparameter multivariate analysis. To appear in J. Multiv. Analysis.

    Google Scholar 

  20. Stahel, W. A. (1981). Breakdown of covariance estimators. Research Report 31, Fachgruppe für Statistik, ETH, Zürich.

    Google Scholar 

  21. Tukey, J. W. (1975). Mathematics and picturing data. Proc. Intern. Congr. Math. Vancouver 1974 2 523–531.

    MathSciNet  Google Scholar 

  22. Zuo, Y. (2000). Projection based depth functions and associated medians. Tentatively accepted by Ann. Statist.

    Google Scholar 

  23. Zuo, Y., Cui, H., and He, X. (2001). On the Stahel-Donoho estimators and depth-weighted means of multivariate data. Tentatively accepted by Ann. Statist.

    Google Scholar 

  24. Zuo, Y., and Serfling, R. (2000a). General notions of statistical depth function. Ann. Statist. 28(2) 461–482.

    Article  MATH  MathSciNet  Google Scholar 

  25. Zuo, Y., and Serfling, R. (2000b). Structural properties and convergence results for contours of sample statistical depth functions. Ann. Statist 28(2) 483–499.

    Article  MATH  MathSciNet  Google Scholar 

  26. Zuo, Y., and Serfling, R. (2000c). On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. J. Statist. Plann. Inference 84 55–79.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yijun Zuo

  2. Department of Mathematics and Statistics, Arizona State University, Tempe, AZ, 85287-1804, USA

    Yijun Zuo

Authors
  1. Yijun Zuo
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Editor information

Editors and Affiliations

  1. Statistics Group, University of Neuchâtel, P.O. Box 805, CH-2002, Neuchâtel, Switzerland

    Yadolah Dodge (Prof. of Statistics and Operation Research) (Prof. of Statistics and Operation Research)

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© 2002 Springer Basel AG

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Zuo, Y. (2002). Multivariate Trimmed Means Based on Data Depth. In: Dodge, Y. (eds) Statistical Data Analysis Based on the L1-Norm and Related Methods. Statistics for Industry and Technology. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8201-9_26

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  • DOI: https://doi.org/10.1007/978-3-0348-8201-9_26

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9472-2

  • Online ISBN: 978-3-0348-8201-9

  • eBook Packages: Springer Book Archive

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