Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 4 / a

The risk set at $x_i$ contains every individual still under observation and event-free immediately before $x_i$. Since the observed times are strictly ordered increasingly, these are individuals $i,i+1,\dots ,n$, so its size is

At time $x_i$, the observed number of events is $v_i$ and the exposure to the common instantaneous hazard is the risk-set size $n-i+1$. The likelihood score for a hazard increment therefore equates observed and expected events:Thus the Nelson–Aalen estimator isIt estimates the cumulative hazard function by adding event count divided by current exposure at every observed event time.