Mixed log-likelihood derivative obstruction to parameter separation (${\partial }_{\psi }{\partial }_{\lambda }\log f$)

On a common positive support, a factorization $f={\varphi }_0(x){\varphi }_C(x;\psi ){\varphi }_S(x;\lambda )$ implies ${\partial }_{\psi }{\partial }_{\lambda }\log f=0$. Thus any nonzero mixed derivative prevents this exact separation. For $n$ independent gamma distributed observations with shape $a$ and rate $b$, this mixed derivative is $n/b$, so exact separation of shape and rate into such factors is impossible.