Distribution-Free Conditional Median Inference

TL;DR

This paper introduces a distribution-free method for constructing confidence intervals for the conditional median of a response variable, ensuring coverage without distribution assumptions, and establishes fundamental limits on interval length.

Contribution

It proposes a conformal prediction-based approach for distribution-free conditional median inference and proves a universal lower bound on interval length.

Findings

Guarantees coverage under any distribution

Sharp performance bounds for specific distributions

A lower bound on interval length independent of sample size

Abstract

We consider the problem of constructing confidence intervals for the median of a response $Y \in \mathbb{R}$ conditional on features $X \in \mathbb{R}^d$ in a situation where we are not willing to make any assumption whatsoever on the underlying distribution of the data $(X,Y)$. We propose a method based upon ideas from conformal prediction and establish a theoretical guarantee of coverage while also going over particular distributions where its performance is sharp. Additionally, we prove an equivalence between confidence intervals for the conditional median and confidence intervals for the response variable, resulting in a lower bound on the length of any possible conditional median confidence interval. This lower bound is independent of sample size and holds for all distributions with no point masses.