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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.

A scale family rescales a fixed distribution by a positive parameter. The standardized variable $Y/\sigma$ has density $f$, independent of $\sigma$. This simplifies moments, simulations and the Jeffreys prior for a scale parameter.