Let $Y(t)$ be the <at-risk process> and $N(t)$ the <counting process> for observed events. Over a short interval, the multiplicative-intensity model gives
$$
\mathbb E\{dN(t)\mid\mathcal F_{t-}\}=Y(t)h(t),dt
=Y(t),dH(t),
$$
where $h$ is the <hazard function> and $H$ the <cumulative hazard function>. Solving this relation for the infinitesimal hazard increment suggests $d\widehat H(t)=dN(t)/Y(t)$. Summing over distinct event times gives the <Nelson–Aalen estimator>
$$
\widehat H(t)=\sum_{j:a_j\leq t}\frac{d_j}{r_j},
$$
where $d_j$ events occur among $r_j$ individuals at risk. Here there are no ties, so $d_j=1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 4 / b / Solution

{title2=$Y(t)$}

An at-risk process indicates or counts which individuals belong to the <risk set> at each time.

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 Incoming links: At-risk process