Properties and robustness of scatter halfspace depth

G. Louvet1 G. Van Bever2 and S. Nagy3
  • 1

    Université libre de Bruxelles, Belgium [gaetan.louvet@ulb.be]

  • 2

    Université libre de Bruxelles, Belgium [germain.van.bever@ulb.be]

  • 3

    Charles University, Czech Republic [nagy@karlin.mff.cuni.cz]

Keywords: Scatter depth – Robustness

Statistical depth provides robust nonparametric tools to analyze distributions. Depth functions indeed measure the adequacy of distributional parameters to underlying probability measures. In the location case, the celebrated halfspace depth (Tukey [1975]) has been widely studied. Recently, depth notions for scatter parameters have been defined (Chen et al. [2018]).

The robustness properties of the Tukey depth have already been amply discussed (Romanazzi [2001], Liu et al. [2017]). However, the robustness of the scatter depth remains largely unknown. In this talk, we present several results regarding the scatter depth function and its associated scatter median, including the influence function and the breakdown point. We also present an alternative depth measure, defined as a symmetrized version of the scatter depth, and explore its improved robustness.

The theoretical properties of the scatter depth have been widely studied (Paindaveine and Van Bever [2018]). In this work, we investigate new properties, mainly based on the scatter median and its minimizing directions. In addition, we study finite sample properties of this depth, examining, for example, the maximum depth value depending on the sample size and dimension.

References

  • Chen et al. [2018] M. Chen, C. Gao, and Z. Ren. Robust covariance and scatter matrix estimation under huber’s contamination model. Ann. Statist., 46(5):1932–1960, 2018.
  • Liu et al. [2017] X. Liu, Y. Zuo, and Q. Wang. Finite sample breakdown point of tukey’s halfspace median. Science China Mathematics, 60(5):861–874, February 2017. ISSN 1869-1862. doi: 10.1007/s11425-016-0285-1. URL http://dx.doi.org/10.1007/s11425-016-0285-1.
  • Paindaveine and Van Bever [2018] D. Paindaveine and G. Van Bever. Halfspace depths for scatter, concentration and shape matrices. Ann. Statist., 46(6B):3276–3307, 2018.
  • Romanazzi [2001] M. Romanazzi. Influence function of halfspace depth. J. Multivariate Anal., 77(1):138–161, 2001. ISSN 0047-259X.
  • Tukey [1975] J. W. Tukey. Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians, pages 523–531, 1975.