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Influence Function of Halfspace Depth

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  • Romanazzi, Mario

Abstract

The sensitivity of halfspace depth values and contours to perturbations of the underlying distribution is investigated. The influence function of the halfspace depth of any point x[set membership, variant]p is bounded and discontinuous; it is constant and positive when the perturbing observation z is placed in any optimal halfspace and it is constant and negative when z is placed in any non-optimal halfspace. When the optimal halfspace is unique a von Mises expansion allows an easy derivation of the asymptotic distribution of the sample halfspace depth. In the sampling case, in general, addition of a single observation outside the convex hull of the sample alters all the depth regions but only the outer region can be arbitrarily expanded. To obtain the same effect on the inner regions the size of the perturbation is required to be not less than the depth orders. Numerical illustrations of the results are given.

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  • Romanazzi, Mario, 2001. "Influence Function of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 138-161, April.
    • Handle: RePEc:eee:jmvana:v:77:y:2001:i:1:p:138-161
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    References listed on IDEAS

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    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.
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    5. Benjamin Avanzi & Mark Lavender & Greg Taylor & Bernard Wong, 2022. "Detection and treatment of outliers for multivariate robust loss reserving," Papers 2203.03874, arXiv.org, revised Jun 2023.

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