# Distribution-free properties of isotonic regression

## TL;DR

This paper reveals distribution-free properties of isotonic regression, showing that the slopes of the greatest convex minorant follow order statistics of running averages, leading to exact risk formulas for constant true sequences.

## Contribution

It establishes that the slopes of the greatest convex minorant are distributed as order statistics of running averages under exchangeable errors, providing exact risk formulas.

## Key findings

- Slopes follow order statistics of running averages
- Exact non-asymptotic risk formula for constant sequences
- Distribution-free properties under exchangeable errors

## Abstract

It is well known that the isotonic least squares estimator is characterized as the derivative of the greatest convex minorant of a random walk. Provided the walk has exchangeable increments, we prove that the slopes of the greatest convex minorant are distributed as order statistics of the running averages. This result implies an exact non-asymptotic formula for the squared error risk of least squares in isotonic regression when the true sequence is constant that holds for every exchangeable error distribution.