Cumulative incidence update with tied censoring ($\Delta {\hat{F}}_j(t)=\hat{S}(t-)d_j(t)/Y(t-)$)

For competing risks, a cumulative incidence function jumps by the displayed Aalen–Johansen estimator increment. A person right-censored at exactly the event time remains in the immediately preceding risk set under the usual events-before-censoring tie convention. Censoring changes the next risk set, not the current event numerator or survival jump.