Aalen–Johansen estimator ($\hat{P}(0,t)={\prod }_{u\le t}(I+d\hat{A}(u))$)

The Aalen–Johansen estimator is a product of empirical state-transition matrices over event times. In competing risks, the estimated cumulative incidence function for cause $j$ jumps by $\hat{S}(t-)d_j(t)/Y(t)$, while overall survival is updated using all event causes. Right censoring changes subsequent risk sets but does not itself create a probability jump.