 Empirical likelihood with mixed censoring

For independent event times with survivor function $S(t)=\mathbb{P}(T>t)$, an exact event at $t$ contributes mass $S(t-)-S(t)$, a right-censored observation at $c$ contributes $S(c)$, and an interval-censored observation $l<T\le u$ contributes $S(l)-S(u)$. Multiplying these factors gives the event-time likelihood when the observation mechanism is noninformative. Maximizing over nonincreasing right-continuous survivor functions gives a nonparametric maximum-likelihood estimator rather than requiring a parametric density.