Regression medians and uniqueness
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Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999), the most famous, exemplifies a direct extension of Tukey location depth (Tukey (1975)) to regression. The extension of another prevailing location depth, the projection depth (Liu (1992), and Zuo and Serfling (2000)), to regression is called the projection regression depth (PRD) (Zuo (2018a)(Z18a)). These two represent the most promising depth notions in regression (Z18a). Carrizosa depth D_C (Carrizosa (1996)) is another notion of depth in regression. The most remarkable advantage of the notion of depth in regression is to introduce directly, the median-type estimator, the maximum (or deepest) regression depth estimator (regression median) for regression parameters in a multi-dimensional setting. The maximum (deepest) regression depth estimators (regression medians) serve as robust alternatives to the classical least squares or least absolute deviations estimator of the unknown parameters in a general linear regression model. The uniqueness of regression medians is central in the discussion of the asymptotics of sample regression medians and a desirable property for the convergence of approximate algorithm in the computation of sample regression medians. Are the regression medians induced from RD, PRD, and D_C unique? Answering this question is the goal of this article. It is found that the regression median induced from PRD possesses the uniqueness property, unlike its leading competitors. This and other findings on the performance of PRD and its induced median indicate that the PRD and its induced median are highly favorable among its leading competitors.
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