Empirical likelihood

Empirical likelihood places probability masses on observed or data-supported values and maximizes their product subject to normalization and any estimating constraints.

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.

Partition the time axis into cells on which membership in each observed exact-event or censoring set is constant. Let $w_j$ be a cell's probability mass and $A_{ij}$ indicate whether cell $j$ is permitted by observation $i$. The empirical likelihood has the displayed form, with nonnegative masses summing to one. Columns of $A$ that agree can be combined. If column $j$ is componentwise dominated by another column $k$, moving all mass from $j$ to $k$ cannot reduce any factor, so an unconstrained nonparametric maximum-likelihood estimator can omit column $j$. The remaining log-likelihood is a concave function of the masses. Extra constraints can prevent a domination move; for a fixed survivor function value, split cells at the constrained time and retain the associated linear mass constraint.

Suppose three exact event times $t_1<t_2<t_3$ coexist with a right-censored time $b$ and an interval $(a,c]$, ordered as $t_1<a<b<t_2<c<t_3$. Set $u=S(a)$ and $v=S(c)$. Monotonicity bounds the three event masses by $1-u$, $u-v$ and $v$, while $S(b)\le u$. Thus the mixed-censoring likelihood is bounded by the displayed expression. Equality is attained by putting all mass at $t_1,t_2,t_3$. The reduced log-likelihood is strictly concave for $0<v<u<1$, and its stationary equations yield $u=4/5$, $v=4/15$. The fitted masses are $1/5$, $8/15$, $4/15$.