Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 218 / 1 / d / Solution

The variance $\mu +{\mu }^2/\theta$ approaches the Poisson distribution variance $\mu$ when $\theta \to \infty$. Equivalently, with $\alpha =1/\theta \ge 0$, the Poisson model is the boundary value $\alpha =0$. The usual Wilks theorem for a likelihood-ratio test assumes that the null parameter is an interior point of a smooth parameter space, so comparing the statistic with an ordinary ${\chi }_1^2$ law is invalid here.

A valid parametric bootstrap proceeds as follows. First fit the null Poisson model and retain its fitted means ${\hat{\mu }}_i$. For each bootstrap repetition $b=1,\dots ,B$, independently simulateusing the original doses, refit both the Poisson and negative-binomial models to that simulated data, and calculateFor the observations calculate the analogous $T_{obs}$. The bootstrap $p$-valueuses the null distribution with its boundary and finite-sample fitting behaviour automatically reproduced. A large $B$ controls the Monte Carlo error of this estimate.