Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 28J / d / Solution

Under the null model, $\pi =1$, which is a boundary point of the parameter space. Standard interior-point asymptotic normality of the maximum-likelihood estimator therefore need not apply.

Indeed, put $a=1-e^{-\lambda }$. Under $\pi =1$, $N_{+}\sim \mathrm{Binomial}(n,a)$. For the unconstrained estimator $\tilde{\pi }=N_{+}/(na)$, the central limit theorem givesSince $\hat{\pi }=\min \{1,\tilde{\pi }\}$, the continuous mapping theorem yieldsThis distribution has probability $1/2$ at zero and a continuous normal half-density on the negative half-line, so it is not normal.