 Solution

Since the observed times are ordered, the final risk set is $R_m=\{m\}$. Its Cox partial likelihood factor is thereforefor either value of the last event indicator. Every earlier risk set and earlier event factor is unaffected by changing $v_m$. Thus the entire partial likelihood as a function of $\beta$ is unchanged, not merely its value at one fitted coefficient. Its maximizing set, its score function and its observed Fisher information are identical. In particular, any consistently chosen finite fitted $\hat{\beta }$ is unchanged. The lone terminal event conveys no relative-hazard information because there is no other individual with whom to compare it.