Graduate Student Seminar: Empowering Quantile Regression Inference with Low-Cost Predictions

We study quantile regression in a data-limited setting where the gold-standard outcome is available only for a limited number of observations, while low-cost predictions are widely available. The growing availability of predictions from modern AI has motivated research on prediction-powered inference, which uses these predictions to improve statistical inference. Naively extending this framework to quantile regression, however, raises some challenges: computational difficulties due to the discontinuity of the subgradient, and overly conservative confidence intervals. To address these issues, we propose a convolution-based smoothing of the check-loss objective and develop two variants of the estimator. The proposed estimators are computationally tractable, and our numerical studies show that they mitigate overcoverage. As a theoretical contribution, we establish the asymptotic distributions of the proposed estimators under a possibly misspecified linear quantile regression model. We further propose an ensemble of the two estimators and illustrate the proposed methods through simulations and an application to a local housing dataset.