Distribution-Free Prediction Sets for Regression under Target Shift

In real-world applications, the limited availability of labeled outcomes presents significant challenges for statistical inference due to high collection costs, technical barriers, and other constraints. In this work, we propose a method to construct efficient conformal prediction sets for new target outcomes by leveraging a source distribution that is distinct from the target but related through a distributional shift assumption and provides abundant labeled data. When the target data are fully unlabeled, our predictions rely solely on the source distribution, whereas partial target labels, when available, are integrated to improve efficiency. To address the challenges of data non-exchangeability and distribution non-identifiability, we identify the likelihood ratio by matching the covariate distributions of the source and target domains within a finite B-spline space. To accommodate complex error structures such as asymmetry and multimodality, our method constructs highest predictive density sets using a novel weight-adjusted conditional density estimator. This estimator models the source conditional density along a quantile process and transforms it, through appropriate weighting adjustments, to approximate the target conditional density. We establish the theoretical properties of the proposed method and evaluate its finite-sample performance through simulation studies and a real-data application to the MIMIC-III clinical database.