Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 6 / c

There are seven deaths by day 28. Their total time plus 28 days from each of the ten later observations gives first-interval exposureAmong the ten observations beyond day 28, eight are deaths, and their exposure beyond 28 is

Under the two-rate alternative, maximize the piecewise-exponential survival model likelihood separately to obtain the estimates in part (i). Under $H_0:{\theta }_1={\theta }_2=\theta$, there are 15 deaths and 2200 days of exposure, so $\hat{\theta }=15/2200$. Twice the maximized log-likelihood difference isUnder the null this likelihood-ratio test statistic is asymptotically ${\chi }_1^2$.