Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 1 / 6H / b / Solution

A sufficient statistic $T$ is minimal sufficient if it is a function of every other sufficient statistic: whenever $U$ is sufficient, there is a function $h$ such that $T=h(U)$ almost surely, up to the usual null-set qualification. Thus its level sets give the coarsest sufficient partition of the sample space. In a dominated family with positive densities, the likelihood-ratio criterion for minimal sufficiency says that $T$ is minimal sufficient when