= Solution
Let $t_j$ be the distinct observed event times, with $r_j$ individuals at risk just before $t_j$ and $d_j$ events there. Parameterize the event-time distribution by the conditional failure probabilities $q_j=\mathbb P(T=t_j\mid T\geq t_j)$. Under <independent censoring>, the survival part of the nonparametric <likelihood> groups into factors
$$
L(q)\propto\prod_j q_j^{d_j}(1-q_j)^{r_j-d_j}.
$$
An individual still under observation at $t_j$ contributes either a failure factor or a survival factor; later-censored individuals contribute the latter factors until censoring. Maximizing each factor independently gives $\widehat q_j=d_j/r_j$, including the boundary solutions when $d_j=0$ or $d_j=r_j$. Multiplying the conditional survival probabilities yields the <Kaplan–Meier estimator>
$$
\boxed{\widehat F(t)=\prod_{t_j\leq t}\left(1-\frac{d_j}{r_j}\right).}
$$
The notation $F$ here denotes the <survivor function>, as in the question. The estimator is constant between event times. Under the usual simultaneous-event convention, individuals censored at $t_j$ remain in its <risk set> while events are processed, and are then removed.
Back to article page
