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Some Extensions of Tukey's Depth Function

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  • Zhang, Jian

Abstract

As the extensions of Tukey's depth, a family of affine invariant depth functions are introduced for multivariate location and dispersion. The location depth functions can be used for the purpose of multivariate ordering. Such kind ordering can retain more information from the original data than that based on Tukey's depth. The dispersion depth functions provide some additional view of the dispersion of the data set. It is shown that these sample depth functions converge to their population versions uniformly on any compact subset of the parameter space. The deepest points of these depth functions are affine equivariant estimates of multivariate location and dispersion. Under some general conditions these estimates are proved to have asymptotic breakdown points at least 1/3 and convergence rates of 1/. Their asymptotic distributions are also obtained under some regularity conditions. A new algorithm based on the idea of thresholding is presented for computing these kinds of estimates and realized in the bivariate case. Simulations indicate that some of them could have the empirical mean squared errors smaller than those based on Tukey's depth function or Donoho's depth function.

Suggested Citation

  • Zhang, Jian, 2002. "Some Extensions of Tukey's Depth Function," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 134-165, July.
    • Handle: RePEc:eee:jmvana:v:82:y:2002:i:1:p:134-165
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    References listed on IDEAS

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    1. Maronna, Ricardo A. & Stahel, Werner A. & Yohai, Victor J., 1992. "Bias-robust estimators of multivariate scatter based on projections," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 141-161, July.
    2. 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. Miguel Arcones & Hengjian Cui & Yijun Zuo, 2006. "Empirical depth processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(1), pages 151-177, June.
    2. Davy Paindaveine & Germain Van Bever, 2017. "Halfspace Depths for Scatter, Concentration and Shape Matrices," Working Papers ECARES ECARES 2017-19, ULB -- Universite Libre de Bruxelles.
    3. Xin Dang & Robert Serfling & Weihua Zhou, 2009. "Influence functions of some depth functions, and application to depth-weighted L-statistics," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(1), pages 49-66.
    4. Davy Paindaveine & Germain Van Bever, 2017. "Tyler Shape Depth," Working Papers ECARES ECARES 2017-29, ULB -- Universite Libre de Bruxelles.

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