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

The variance of an estimated hazard increment is large when its group has few individuals in the risk set. The log-rank weights are near zero when either $r_j^{(0)}$ or $r_j^{(1)}$ is small and are largest when both groups retain substantial information. They therefore suppress noisy late-event comparisons and weight each observed-minus-expected event by its available information. Unit weights would instead give equal influence to unstable increments from depleted risk sets.