 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 $q_j=\mathbb{P}(T=t_j\mid T\ge t_j)$. Under independent censoring, the survival part of the nonparametric likelihood groups into factorsAn 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 ${\hat{q}}_j=d_j/r_j$, including the boundary solutions when $d_j=0$ or $d_j=r_j$. Multiplying the conditional survival probabilities yields the Kaplan–Meier estimatorThe 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.