= 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
$$
\boxed{T(x)=T(y)
\quad\Longleftrightarrow\quad
\frac{p_\theta(x)}{p_\theta(y)}
\text{ is independent of }\theta.}
$$
Back to article page
