Solution
= Solution
Write $X=\sigma Z$ with $Z\sim F_1$. Then
$$
Y=\log X^2=\log\sigma^2+\log Z^2=\theta+W,
$$
where the distribution of $W=\log Z^2$ does not depend on $\sigma$. Consequently
$$
G_\theta(y)=G_0(y-\theta),
$$
which 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.