 Solution

Assume the scale family has the differentiability needed for Fisher information, and the following constant is finite and positive. With ${\ell }_f(u)=\log f(u)$, the score function isPart (b) therefore giveswhere $C$ is independent of $\sigma$. The Jeffreys prior for a scale parameter is consequentlyThe reciprocal-of-a-reciprocal in the TeX is a transcription error; the PDF has ${\sigma }^{-1}$. The expected-Hessian information identity should not be applied indiscriminately to nonregular parameter-dependent supports.