Prediction Sets and Conformal Inference with Interval Outcomes

Given data on a random variable $Y$, a prediction set with miscoverage level $\alpha \in (0,1)$ is a set that contains a new draw of $Y$ with probability $1-\alpha$. Among all prediction sets satisfying this coverage property, the oracle prediction set is the one with minimal volume. The oracle prediction set offers a complementary view of the distribution of $Y$, beyond point estimators such as the mean and quantiles, and has attracted considerable interest recently. This paper develops methods for estimating such prediction sets conditional on observed covariates when $Y$ is \textit{censored} or \textit{interval-valued}. We characterise the oracle prediction set under partial identification induced by interval censoring and propose consistent estimators for both oracle prediction intervals and more general oracle prediction sets consisting of multiple disjoint intervals. In addition, we apply conformal inference to construct finite-sample valid prediction sets for interval outcomes that remain consistent as the sample size grows, using a conformity score tailored to interval data. The proposed procedure accounts for irreducible prediction uncertainty due to the stochastic nature of outcomes, modelling uncertainty arising from partial identification, and sampling uncertainty that vanishes as sample size increases. We conduct Monte Carlo simulations and two empirical applications using UK job postings data and the US Current Population Survey. The results demonstrate the robustness and efficiency of the proposed methods.