 Solution

The history ${\mathcal{H}}_{t-}$ contains the information observed strictly before $t$: baseline characteristics, previous events and censoring, and therefore which individuals are currently eligible and under observation. Formally it is the pre-$t$ information in the relevant filtration. The at-risk process $Y_i(t)$ is 1 if individual $i$ is observed and event-free immediately before $t$, and 0 otherwise. Let $Y_{+}(t)={\sum }_i{Y}_i(t)$ and write $dH(t)=h(t)dt$ for the cumulative hazard increment.

The counting-process intensity in survival analysis givesThe common conditional hazard function is assumed to remain applicable after conditioning on the observed history, as under suitable independent censoring. Summing gives the conditional mean of the total event increment:On times with $Y_{+}(t)>0$, invert this relation to estimate the hazard increment:The last sum is for untied events. This is the Nelson–Aalen estimator; for ties replace 1 by the number of events at that time. No increment can be estimated once the risk set is empty.