Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 5 / c / Solution

Put $D_k={\sum }_{i:z_i=k}{d}_i$ and $T_k={\sum }_{i:z_i=k}{x}_i$. Since the cumulative hazard is $\theta e^{z_i\beta }{x}_i$,The score equations imply $D_1=\theta e^{\beta }{T}_1$ and $D_0=\theta T_0$. Thereforewhen both event counts are positive, with the usual infinite boundary estimates otherwise.