Minimal sufficient statistic

A sufficient statistic is minimal sufficient when it is a function of every other sufficient statistic, up to null sets. Its level sets form the coarsest sufficient partition of the sample space.

For independent samples uniform on $[-\theta ,\theta ]$ with $\theta >0$, the absolute maximum $M$ is a minimal sufficient statistic, since the likelihood is $(2\theta {)}^{-n}{1}_{\{\theta \ge M\}}$. The pair of sample minimum and maximum is a sufficient statistic but is not minimal: simultaneous sign reversal preserves $M$ and generally changes that pair.

For a dominated family with positive densities $p_{\theta }$, a statistic $T$ is minimal sufficient if