# A note on breaking ties among sample medians

Source: [A note on breaking ties among sample medians](<https://www.sota2.com/research/paper/68fa7dc51e209131cea945f6>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-26T06:26:24.669Z

Authors: Peter M. Aronow, Donald K. K. Lee

Published: 2018-07-10T03:23:06.000Z

## Abstract

Given samples $x\_1,\\cdots,x\_n$, it is well known that any sample median value (not necessarily unique) minimizes the absolute loss $\\sum\_{i=1}^n \|q-x\_i\|$. Interestingly, we show that the minimizer of the loss $\\sum\_{i=1}^n\|q-x\_i\|^{1+\\epsilon}$ exhibits a singular perturbation behaviour that provides a unique definition for the sample median as $\\epsilon \\rightarrow 0$. This definition is the unique point among all candidate median values that balances the $logarithmic$ moment of the empirical distribution. The result generalizes directly to breaking ties among sample quantiles when the quantile regression loss is modified in the same way.

## Recorded benchmark results

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## Original sources and implementation

[Original paper](<https://arxiv.org/abs/1807.03462>)

[PDF](<https://arxiv.org/pdf/1807.03462.pdf>)