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.
Highlights
Regular univariate sample median defined as the innermost point of a data set is unique (If the sample median is defined to be the point θ that minimizes the sum of its distances to sample points (i.e., θ = arg minθ ∈R1 ∑in=1 |θ − xi |, where xi, i = 1, · · ·, n are the given n sample points in R1 ), it is not unique
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 Regression depth (RD) and DC cases. These and other findings indicate that the projection regression depth (PRD) and its induced median are highly favorable among their leading competitors
Robustness of the median induced from RD and PRD have been investigated in Van Aelst and Rousseeuw (2000) (VAR00) [13] and Zuo (2018b) [14], respectively
Summary
One of the outstanding advantages of depth notions is that they can be directly employed to introduce median-type deepest estimating functionals (or estimators in the empirical case) for the location or regression parameters in a multi-dimensional setting based on a general min-max stratagem. Robustness of the median induced from RD and PRD have been investigated in Van Aelst and Rousseeuw (2000) (VAR00) [13] and Zuo (2018b) [14], respectively These medians, just like their location or univariate counterpart, possess high breakdown point robustness. Note that β∗ might not be unique, and a conventional remedy measure is to take the average of all maximum depth points This could lead to a scenario where the resulting functional (or estimator) might not have the maximum depth any more.
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