= Conditional scale density in a curved Gaussian family
{title2=$f(v\mid a;\mu)\propto\mu^{-n}v^{n-1}e^{-nq^2v^2/(2\mu^2)+nqv/(a\mu)}$}
For <independent> $N(\mu,\mu^2)$ observations with $\mu>0$, the <sufficient statistics> are their sum and squared norm. Condition on $a=\sqrt n\,\|y\|/\sum_i y_i>0$ and write $v=\widehat\mu=\|y\|/(q\sqrt n)$, where $q=(\sqrt{1+4a^2}+1)/(2a)$. <Polar coordinates> supply radial measure $\|y\|^{n-1}d\|y\|$, while the conditional angular factor does not depend on $\mu$. This proves the displayed shape. Its normalization depends on $a,n$ but not $\mu$ after the substitution $v=\mu z$, and agrees with the normalized <P-star approximation>.
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