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Depth Induced Regression Medians and Uniqueness

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  • Yijun Zuo

    (Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA)

Abstract

The notion of median in one dimension is a foundational element in nonparametric statistics. It has been extended to multi-dimensional cases both in location and in regression via notions of data depth. Regression depth (RD) and projection regression depth (PRD) represent the two most promising notions in regression. Carrizosa depth D C is another depth notion in regression. Depth-induced regression medians (maximum depth estimators) serve as robust alternatives to the classical least squares estimator. The uniqueness of regression medians is indispensable in the discussion of their properties and the asymptotics (consistency and limiting distribution) of sample regression medians. Are the regression medians induced from RD, PRD, and D C unique? Answering this question is the main goal of this article. It is found that only the regression median induced from PRD possesses the desired uniqueness property. The conventional remedy measure for non-uniqueness, taking average of all medians, might yield an estimator that no longer possesses the maximum depth in both RD and D C cases. These and other findings indicate that the PRD and its induced median are highly favorable among their leading competitors.

Suggested Citation

  • Yijun Zuo, 2020. "Depth Induced Regression Medians and Uniqueness," Stats, MDPI, vol. 3(2), pages 1-13, April.
    • Handle: RePEc:gam:jstats:v:3:y:2020:i:2:p:9-106:d:343918
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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. Carrizosa, Emilio, 1996. "A Characterization of Halfspace Depth," Journal of Multivariate Analysis, Elsevier, vol. 58(1), pages 21-26, July.
    3. Zuo, Yijun, 2013. "Multidimensional medians and uniqueness," Computational Statistics & Data Analysis, Elsevier, vol. 66(C), pages 82-88.
    4. 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.
    5. Van Aelst, Stefan & Rousseeuw, Peter J. & Hubert, Mia & Struyf, Anja, 2002. "The Deepest Regression Method," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 138-166, April.
    6. Mingxin Wu & Yijun Zuo, 2008. "Trimmed and winsorized standard deviations based on a scaled deviation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(4), pages 319-335.
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