= Mixed log-likelihood derivative obstruction to parameter separation
{title2=$\partial_\psi\partial_\lambda\log f$}
On a common positive support, a factorization $f=\phi_0(x)\phi_C(x;\psi)\phi_S(x;\lambda)$ 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.
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