# Distribution-Free Conditional Median Inference

arXiv: https://arxiv.org/abs/2102.07967 · HTML: https://mlsanity.com/paper/2102.07967/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Feb 2021.

## Built on by (most cited first, 7 of 7)

- [Improving Conditional Coverage via Orthogonal Quantile Regression](https://mlsanity.com/paper/2106.00394/) (2021)
- [Posterior-Variance-Based Error Quantification for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.12499/) (2022)
- [Conformal Frequency Estimation using Discrete Sketched Data with Coverage for Distinct Queries](https://mlsanity.com/paper/2211.04612/) (2022)
- [Conformal Frequency Estimation with Sketched Data](https://mlsanity.com/paper/2204.04270/) (2022)
- [The Limits of Assumption-free Tests for Algorithm Performance](https://mlsanity.com/paper/2402.07388/) (2024)
- [Are all models wrong? Fundamental limits in distribution-free empirical model falsification](https://mlsanity.com/paper/2502.06765/) (2025)
- [L1 Estimation: On the Optimality of Linear Estimators](https://mlsanity.com/paper/2309.09129/) (2023)

## Abstract (arXiv)

We consider the problem of constructing confidence intervals for the median of a response Y ∈ ℝ conditional on features X ∈ ℝd 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.