Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 5 / a / Solution

Write $\theta =e^{\beta }$ for the group-1 to group-0 hazard ratio. At $t_1$, all four subjects are at risk, so subject 1 contributes $1/(2+2\theta )$. At $t_2$, subjects 2, 3, and 4 are at risk, so subject 2 contributes $\theta /(1+2\theta )$.

If $t_2<c<t_4$, subject 3 leaves before $t_4$ and subject 4, if it fails, is alone in its risk set. Henceindependently of $k$. Its log derivative vanishes when $1-2{\theta }^2=0$, giving

If $c=t_4$, subject 3 remains in the risk set at a failure of subject 4. ThereforeFor $k=0$, $\hat{\theta }=1/\sqrt{2}$. For $k=1$, the score equation is $1-\theta -4{\theta }^2=0$, soThus $\hat{\theta }$ is constant at $1/\sqrt{2}$ for $t_2\le c<t_4$. At $c=t_4$ it remains there when $k=0$ and drops to approximately $0.39$ when $k=1$, because only then does the censoring time change an event's risk set.