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Finite sample valid probabilistic inference on quantile regression

In most applications, uncertainty quantification in quantile regression take the form of set estimates for the regression coefficients. However, often a more informative type of uncertainty quantification is desired where other inference-related tasks can be performed, such as the assignment of (imprecise) probabilities to assertions of interest about (any feature of) the regression coefficients. Validity of these probabilities, in the sense that their values are well-calibrated in a frequentist sense, is fundamental to the trustworthiness of the drawn conclusions. This paper presents a nonparametric Inferential Model (IM) construction that offers provably valid probabilistic uncertainty quantification in quantile regression, even in finite sample settings. It is also shown that this IM can be used to derive finite sample confidence regions for (any feature of) the regression coefficients. As a result, regardless of the type of uncertainty quantification desired, the proposed IM offers an appealing solution to quantile regression problems.