Theorem: Let $p(\x \mid \theta)$ denote the joint pdf or pmf of a sample $\x$. Suppose there exists a function $T(\x)$ such that $\forall\,\x$ and $\y$, the ratio $p(\x \mid \theta)/p(\y \mid \theta)$ is a constant function of $\theta$ if and only if $T(\x)=T(\y)$. Then $T(\x)$ is a minimal sufficient statistic for $\theta$.