Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 4 / 17H / a

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