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Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions

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  • Zuo, Yijun
  • Serfling, Robert

Abstract

Nonparametric notions of multivariate "scatter measure" and "more scattered," based on statistical depth functions, are investigated. In particular, notions of "more scattered" based on the "halfspace" depth function are shown to generalize versions introduced by P. J. Bickel and E. L. Lehmann (1976, 1979) in the univariate case and by M. L. Eaton (1982) and H. Oja (1983) in the multivariate case. Scatter measures are also discussed, with emphasis on those based on the halfspace depth. Basic desirable properties established for the previous versions of "more scattered" are shown to carry over to the depth-based notions as well, in both the univariate and the multivariate cases. Further, some properties unique to the depth-based notions are established.

Suggested Citation

  • Zuo, Yijun & Serfling, Robert, 2000. "Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions," Journal of Multivariate Analysis, Elsevier, vol. 75(1), pages 62-78, October.
    • Handle: RePEc:eee:jmvana:v:75:y:2000:i:1:p:62-78
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    References listed on IDEAS

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    1. Ruts, Ida & Rousseeuw, Peter J., 1996. "Computing depth contours of bivariate point clouds," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 153-168, November.
    2. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    3. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
    4. Nolan, D., 1992. "Asymptotics for multivariate trimming," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 157-169, August.
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    Cited by:

    1. Zuo, Yijun, 2003. "Finite sample tail behavior of multivariate location estimators," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 91-105, April.
    2. Romanazzi, Mario, 2009. "Data depth, random simplices and multivariate dispersion," Statistics & Probability Letters, Elsevier, vol. 79(12), pages 1473-1479, June.
    3. Hwang, Jinsoo & Jorn, Hongsuk & Kim, Jeankyung, 2004. "On the performance of bivariate robust location estimators under contamination," Computational Statistics & Data Analysis, Elsevier, vol. 44(4), pages 587-601, January.

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