Tight Distribution-Free Confidence Intervals for Local Quantile Regression

TL;DR

This paper introduces new distribution-free methods for constructing valid confidence intervals for local quantiles in regression, achieving high accuracy and finite-sample validity without strong distributional assumptions.

Contribution

It proposes the Weighted Quantile and Quantile Rejection methods for distribution-free confidence intervals in local quantile regression, with proven asymptotic and finite-sample guarantees.

Findings

Weighted Quantile achieves nominal coverage with small effective sample sizes (10-20).

Both methods are validated through extensive numerical studies.

The framework allows for confidence intervals at various localization levels.

Abstract

It is well known that it is impossible to construct useful confidence intervals (CIs) about the mean or median of a response $Y$ conditional on features $X = x$ without making strong assumptions about the joint distribution of $X$ and $Y$. This paper introduces a new framework for reasoning about problems of this kind by casting the conditional problem at different levels of resolution, ranging from coarse to fine localization. In each of these problems, we consider local quantiles defined as the marginal quantiles of $Y$ when $(X,Y)$ is resampled in such a way that samples $X$ near $x$ are up-weighted while the conditional distribution $Y \mid X$ does not change. We then introduce the Weighted Quantile method, which asymptotically produces the uniformly most accurate confidence intervals for these local quantiles no matter the (unknown) underlying distribution. Another method, namely, the Quantile Rejection method, achieves finite sample validity under no assumption whatsoever. We conduct extensive numerical studies demonstrating that both of these methods are valid. In particular, we show that the Weighted Quantile procedure achieves nominal coverage as soon as the effective sample size is in the range of 10 to 20.