Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 7 / c / Solution

Let $N(t)={\sum }_i{N}_i(t)$ and $Y(t)={\sum }_i{Y}_i(t)$ be the total observed counting process and size of the risk set. When the individual cumulative hazard functions are all $H_0$, summing the conditional intensity equations givesReplace the expected increment by its observed value and divide by the predictable risk set size. Thus the Nelson–Aalen estimator in integral form iswith the integrand defined as zero when $Y(u)=0$. This is an integral against the step counting process, so each event time contributes its event count divided by the risk set just before that time. The Nelson–Aalen estimator estimates integrated hazard over times with observable risk; it does not supply information after observation has ended.