A note on breaking ties among sample medians

TL;DR

This paper explores a novel way to uniquely define sample medians and quantiles by perturbing the loss function, revealing a singular behavior that balances the empirical distribution's logarithmic moments.

Contribution

It introduces a new method to break ties among sample medians and quantiles using a perturbed loss function, providing a unique and well-defined solution.

Findings

The minimizer of the perturbed loss exhibits singular perturbation behavior.

The new definition balances the logarithmic moment of the empirical distribution.

The approach generalizes to sample quantiles in quantile regression.

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.