Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 41 / 4 / b

Assume the usual cohort setup: all individuals enter at time zero, each contributes one event or censoring time, and every such observation is included. Order these distinct observation times as $t_1<\cdots <t_m$. The risk set size just before $t_k$ is then $r_k=m-k+1$. Write ${\delta }_k=1$ for an event and zero for censoring. Including the jump at an event time, the Nelson–Aalen estimator isExchange the two finite sums:Thus the sum equals the number of observed events. Each event increment appears once for each individual still at risk at that event, exactly cancelling its denominator. This is the event-count identity for Nelson–Aalen cumulative hazards. The cohort and entry convention matter: delayed entry would invalidate the simple equation $r_k=m-k+1$ and require a modified identity.