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Empirical likelihood assigns unknown probability masses $p_j\ge 0$ to data-supported event times or intervals, imposes ${\sum }_j{p}_j=1$, and maximizes the product of each observation's probability. For event-time data, write $F(t)=\mathbb{P}(T\le t)$ and $S(t)=1-F(t)$ for the survivor function. Exact, right-censored, left-censored, interval-censored, and truncated observations contribute the probability of their respective compatible sets.

The maximization uses that $S$ is nonincreasing and right-continuous, $S(0)=1$, and $S(t)\to 0$ as $t\to \infty$. Probability mass need only be placed at endpoints that change an observation's compatible set; moving mass within any observationally indistinguishable interval leaves the likelihood unchanged. Maximizing over those masses gives the nonparametric maximum-likelihood estimator of the survivor function.