Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 17H (Statistics)

A statistic is sufficient when the conditional law of the sample given it is parameter-free; it is minimal sufficient when it is a function of every sufficient statistic. The likelihood factors asso the factorization theorem makes $(n_0,S)$ sufficient. The likelihood-ratio criterion shows that the ratio for two samples is parameter-independent exactly when both $n_0$ and $S$ agree. Thus

For known ${\lambda }_0$, the likelihood ratio of $\pi =1/2$ to $\pi =0$ iswhich increases with $n_0$. Reject for $n_0>c$, with randomization at $n_0=c$ chosen so that the null rejection probability is exactly $\alpha$. Since the zero indicators form a Bernoulli family with success probability increasing in $\pi$ and have a monotone likelihood ratio in $n_0$, the Karlin-Rubin theorem makes this test uniformly most powerful for $\pi >0$.

The null value $\pi =1/2$ is interior and the model is regular. Wilks theorem therefore givesunder the null, because the alternative adds one free parameter. An asymptotic size-$\alpha$ test rejects when