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

Let $N(t)$ count observed events and $Y(t)$ count individuals still in the risk set just before $t$. Under a common hazard function $h(t)$ and independent censoring, the expected event count over a short interval, conditional on the observed past, isHere $H(t)={\int }_0^{t}h(u)du$ is the integrated hazard. Thus an estimating increment for $dH$ is $dN/Y$, on the range where the risk set is nonempty. Accumulating these increments gives the Nelson–Aalen estimatorThe observation times $t_j$ include both events and censorings, $r_j=Y(t_j)$, and ${\delta }_j$ is one for an event and zero for censoring. Without ties each event adds $1/r_j$; censoring changes later risk sets but adds nothing immediately. Equivalently, the local event fraction estimates the local hazard increment, and summing those estimates produces the integrated hazard. The counting-process intensity in survival analysis justifies this estimating equation; it does not imply exact finite-sample unbiasedness after the risk set can become empty.