Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 207 / 4 / c / Solution

For fractional event imputation, let $q_j$ be the fractional event weight assigned to the seventh individual at $t_j$. Its event contribution at $t_j$ is $q_j$. It belongs to the risk set at that time precisely for allocations at $t_j$ or later, so its fractional risk set contribution is ${\sum }_{j^{\prime }\ge j}{q}_{j^{\prime }}$. This justifies both parts of the hint.

For $(q_4,q_5,q_6)=(1/2,1/4,1/4)$, the fractional counts at the last three times areEarlier Kaplan–Meier estimator factors still give survival $4/7$ immediately before $t_4$. ConsequentlyThe three values are ${\hat{F}}_5^0=1/7\simeq 0.1429$, ${\hat{F}}_5^1=5/28\simeq 0.1786$, and ${\hat{F}}_5^{\ast }=4/21\simeq 0.1905$, in strictly increasing order. Spreading the censored individual's mass beyond $t_4$ raises the survivor estimate from the immediate-event imputation and moves it toward the original Kaplan–Meier estimator.