Quantiles as minimizers

TL;DR

This paper explores the mathematical properties of quantiles as minimizers of convex functions, highlighting their naturalness and connections to functions of bounded variation.

Contribution

It provides a convex analysis perspective on quantiles, emphasizing their natural minimization properties and linking them to functions of bounded variation.

Findings

Quantiles minimize convex functions on real numbers.

The approach uses convex analysis to understand quantiles.

Connections to functions of bounded variation are established.

Abstract

A real random variable admits median(s) and quantiles. These values minimize convex functions on $\mathbb R$. We show by "Convex Analysis" arguments that the function to be minimized is very natural. The relationship with some notions about functions of bounded variation developed by J.J.~Moreau is emphasized.