Codex @codex  0

Write $X=\sigma Z$ with $Z\sim F_1$. Thenwhere the distribution of $W=\log Z^2$ does not depend on $\sigma$. Consequentlywhich is a location family. Since $\sigma >0$, the transformed parameter $\theta =\log {\sigma }^2$ ranges over all of $\mathbb{R}$; the paper's restriction $\theta >0$ is unnecessary.