Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 207 / 6 / b / vi / Solution

The second derivative isIt is negative when $v_{+}>0$, verifying a strict maximum. The Observed Fisher information is $v_{+}/{\theta }^2$: more observed side-effects sharpen the likelihood, while the curvature vanishes when no side-effect is observed and the maximum then lies at the boundary $\hat{\theta }=0$.