Efficient Inference for Distributional Treatment Effects with Partially Observed Outcomes

We study the distributional treatment effect when a surrogate outcome is fully observed, but the primary outcome is observed only for a subsample. This setting arises naturally in model or agent evaluation, where automated judges make it inexpensive to score every high-dimensional response while human ratings are often available for only a small subsample. When evaluation scores are discrete or ordinal, a difference between cumulative distribution functions (CDFs) directly measures changes in the probability of low-quality outputs at each rating cutpoint. Building on prediction-powered inference and surrogate-assisted treatment-effect estimation, we develop a cross-fitted estimator that combines conditional CDFs with and without surrogate information. We establish a Gaussian-process limit and uniform confidence bands over outcome thresholds, identify the semiparametric efficiency bound, and characterize variance gains over estimators that reweight the labeled subsample by its known sampling probabilities or adjust for covariates without the surrogate. The construction does not require continuous output density or agreement between the surrogate and primary-outcome scales. Through simulation results and four real data applications, we demonstrate the effectiveness of our approach.