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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 223 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-223/1/b/solution