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Halfspace Depths for Scatter, Concentration and Shape Matrices

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  • Davy Paindaveine
  • Germain Van Bever

Abstract

We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept extends the one from Chen, Gao and Ren (2015) to the non-centered case, and is in the same spirit as the one in Zhang (2002). Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth regions. We do so under minimal assumptions and, in particular, we do not restrict to elliptical distributions nor to absolutely continuous distributions. Interestingly, fully understanding scatter halfspace depth requires considering different geometries/topologies on the space of scatter matrices. We also discuss, in the spirit of Zuo and Serfling (2000), the structural properties a scatter depth should satisfy, and investigate whether or not these are met by the proposed depth. As mentioned above, companion concepts of depth for concentration matrices and shape matrices are also proposed and studied. We illustrate the practical relevance of the proposed concepts by considering a real-data example from finance.

Suggested Citation

  • 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.
  • Handle: RePEc:eca:wpaper:2013/250239
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    References listed on IDEAS

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    Cited by:

    1. Chau, Van Vinh & Ombao, Hernando & von Sachs, Rainer, 2017. "Data depth and rank-based tests for covariance and spectral density matrices," LIDAM Discussion Papers ISBA 2017019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Nagy, Stanislav, 2019. "Scatter halfspace depth for K-symmetric distributions," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 171-177.

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