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Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods

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  • Mizera, Ivan
  • Volauf, Milos

Abstract

Continuity of procedures based on the halfspace (Tukey) depth (location and regression setting) is investigated in the framework of continuity concepts from set-valued analysis. Investigated procedures are depth contours (upper level sets) and maximum depth estimators. Continuity is studied both as the pointwise continuity of data-analytic functions, and the weak continuity of statistical functionals--the latter having relevance for qualitative robustness. After a real-data example, some general criteria and counterexamples are given, as well as positive results holding for "typical" data. Finally, some consequences for diagnostics and practical use of the depth-based techniques are drawn.

Suggested Citation

  • Mizera, Ivan & Volauf, Milos, 2002. "Continuity of Halfspace Depth Contours and Maximum Depth Estimators: Diagnostics of Depth-Related Methods," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 365-388, November.
    • Handle: RePEc:eee:jmvana:v:83:y:2002:i:2:p:365-388
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    References listed on IDEAS

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    1. Van Aelst, Stefan & Rousseeuw, Peter J., 2000. "Robustness of Deepest Regression," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 82-106, April.
    2. Tore Jonsbråten & Roger Wets & David Woodruff, 1998. "A class of stochastic programs withdecision dependent random elements," Annals of Operations Research, Springer, vol. 82(0), pages 83-106, August.
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    1. Mia Hubert & Peter Rousseeuw & Pieter Segaert, 2015. "Multivariate functional outlier detection," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(2), pages 177-202, July.
    2. Yijun Zuo, 2020. "Depth Induced Regression Medians and Uniqueness," Stats, MDPI, vol. 3(2), pages 1-13, April.
    3. Petra Laketa & Stanislav Nagy, 2022. "Halfspace depth for general measures: the ray basis theorem and its consequences," Statistical Papers, Springer, vol. 63(3), pages 849-883, June.
    4. López Pintado, Sara, 2006. "Depth-based inference for functional data," DES - Working Papers. Statistics and Econometrics. WS ws063113, Universidad Carlos III de Madrid. Departamento de Estadística.
    5. Kong, Linglong & Zuo, Yijun, 2010. "Smooth depth contours characterize the underlying distribution," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2222-2226, October.
    6. Lopez-Pintado, Sara & Romo, Juan, 2007. "Depth-based inference for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4957-4968, June.
    7. Zuo, Yijun, 2021. "Computation of projection regression depth and its induced median," Computational Statistics & Data Analysis, Elsevier, vol. 158(C).
    8. Laketa, Petra & Nagy, Stanislav, 2021. "Reconstruction of atomic measures from their halfspace depth," Journal of Multivariate Analysis, Elsevier, vol. 183(C).

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