Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 5 / a / Solution

Define the event counting process and at-risk process byUnder a common hazard function and independent censoring, the conditional event rate is $Y(u)h(u)$. Equivalently, the counting-process compensator satisfies $\mathbb{E}[dN(u)\mid {\mathcal{F}}_{u-}]=Y(u)dH(u)$. Estimate a cumulative hazard increment by the observed event increment divided by the current risk set:With no ties, $dN$ has mass $v_i$ at $x_i$, and $Y(x_i)={\sum }_{j:x_j\ge x_i}1$. Thus the Nelson–Aalen estimator isOnly event times contribute, so no empty risk set is divided into an event count. Right-censored observations contribute time to the risk sets, but no event increment.