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# Depth Induced Regression Medians and Uniqueness

Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA
Stats 2020, 3(2), 94-106; https://doi.org/10.3390/stats3020009
Received: 14 February 2020 / Revised: 16 March 2020 / Accepted: 31 March 2020 / Published: 10 April 2020
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. View Full-Text
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Zuo, Y. Depth Induced Regression Medians and Uniqueness. Stats 2020, 3, 94-106.