# Depth induced regression medians and uniqueness

## TL;DR

This paper investigates the uniqueness of regression medians induced by various data depth notions, finding that only the projection regression depth (PRD) median is uniquely defined, which has implications for robust regression estimation.

## Contribution

It establishes the uniqueness of the regression median induced by PRD and highlights issues with non-uniqueness and averaging methods in other depth notions.

## Key findings

- PRD-induced regression median is unique.
- Averaging medians can lose maximum depth property.
- PRD median is a highly favorable robust estimator.

## Abstract

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.