We identify the distributional and quantile marginal treatment effect functions when the outcome is right-censored. Our method requires a conditionally exogenous instrument and random censoring. We propose asymptotically consistent semi-parametric estimators and valid inferential procedures for the target functions . To illustrate, we evaluate the effect of alternative sentences (fines and community service vs. no punishment) on recidivism in Brazil. Our results highlight substantial treatment effect heterogeneity: we find that people whom most judges would punish take longer to recidivate, while people who would be punished only by strict judges recidivate at an earlier date than if they were not punished.

Predictive algorithms inform consequential decisions in settings with selective labels: outcomes are observed only for units selected by past decision makers. This creates an identification problem under unobserved confounding — when selected and unselected units differ in unobserved ways that affect outcomes. We propose a framework for robust design and evaluation of predictive algorithms that bounds how much outcomes may differ between selected and unselected units with the same observed characteristics. These bounds formalize common empirical strategies including proxy outcomes and instrumental variables. Our estimators work across bounding strategies and performance measures such as conditional likelihoods, mean squared error, and true/false positive rates. Using administrative data from a large Australian financial institution, we show that varying confounding assumptions substantially affects credit risk predictions and fairness evaluations across income groups.