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Integrated rank-weighted depth

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  • Ramsay, Kelly
  • Durocher, Stéphane
  • Leblanc, Alexandre

Abstract

We study depth measures for multivariate data defined by integrating univariate depth measures, specifically, integrated dual (ID) depth introduced by Cuevas and Fraiman (2009) which integrates univariate simplicial depth, and integrated rank-weighted (IRW) depth, which integrates univariate Tukey depth. We build on the results of Cuevas and Fraiman (2009) to show that IRW depth shares many depth properties with ID depth. Further, we provide additional results on exact computation, decreasing along rays, continuity and breakdown point that apply to both ID and IRW depth. We also establish asymptotic normality and consistency of the sample IRW depths. Lastly, we demonstrate the use of this depth measure with real and simulated datasets: calculating robust location estimators and dd-plots.

Suggested Citation

  • Ramsay, Kelly & Durocher, Stéphane & Leblanc, Alexandre, 2019. "Integrated rank-weighted depth," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 51-69.
    • Handle: RePEc:eee:jmvana:v:173:y:2019:i:c:p:51-69
    DOI: 10.1016/j.jmva.2019.02.001
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    References listed on IDEAS

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    1. López-Pintado, Sara & Romo, Juan, 2009. "On the Concept of Depth for Functional Data," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 718-734.
    2. Zhang, Jie & Pan, Meng, 2016. "A high-dimension two-sample test for the mean using cluster subspaces," Computational Statistics & Data Analysis, Elsevier, vol. 97(C), pages 87-97.
    3. Robert Serfling, 2010. "Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 915-936.
    4. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    5. Xiaohui Liu, 2017. "Fast implementation of the Tukey depth," Computational Statistics, Springer, vol. 32(4), pages 1395-1410, December.
    6. Cuevas, Antonio & Fraiman, Ricardo, 2009. "On depth measures and dual statistics. A methodology for dealing with general data," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 753-766, April.
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    Cited by:

    1. Ramsay, Kelly & Durocher, Stephane & Leblanc, Alexandre, 2021. "Robustness and asymptotics of the projection median," Journal of Multivariate Analysis, Elsevier, vol. 181(C).

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