 Solution

For independent exponential survival times with total observed person-time at risk $T_{\bullet }$ and $d$ events, the maximum-likelihood rate is $\hat{h}=d/T_{\bullet }$. A two-group exponential model is a proportional-hazards model because both hazards are constant, so their ratio is constant.

HereThus the group-1 to group-0 hazard-ratio estimate isIt decreases continuously as $c$ increases. The parametric exponential likelihood uses exact exposure times through each arm's person-time, whereas the Cox partial likelihood uses only which subjects belong to each event's risk set. For $t_2<c<t_4$, changing $t_4$ alters group 0 person-time and hence $\hat{\lambda }$, but subject 4 is alone if it fails at $t_4$, so that event contributes one to the partial likelihood and $\hat{\theta }$ is unchanged.