Distribution-free Inferential Models: Achieving finite-sample valid probabilistic inference, with emphasis on quantile regression

This paper presents a novel distribution-free Inferential Model (IM) construction that provides valid probabilistic inference across a broad spectrum of distribution-free problems, even in finite sample settings. More specifically, the proposed IM has the capability to assign (imprecise) probabilities to assertions of interest about any feature of the unknown quantities under examination, and these probabilities are well-calibrated in a frequentist sense. It is also shown that finite-sample confidence regions can be derived from the IM for any such features. Particular emphasis is placed on quantile regression, a domain where uncertainty quantification often takes the form of set estimates for the regression coefficients in applications. Within this context, the IM facilitates the acquisition of these set estimates, ensuring they are finite-sample confidence regions. It also enables the provision of finite-sample valid probabilistic assignments for any assertions of interest about the regression coefficients. As a result, regardless of the type of uncertainty quantification desired, the proposed framework offers an appealing solution to quantile regression.