Theorem: Let \(p(\x|\theta)\) 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 \(p(\x|\theta)/p(\y|\theta)\) is a constant function of \(\theta\) if and only if \(T(\x)=T(\y)\). Then \(T(\x)\) is a minimal sufficient statistic for \(\theta\).