= Orthogonal shape and scale for a generalized Pareto distribution
{title2=$\eta=\sigma(1+\psi)$}
For the positive-shape <generalized Pareto distribution> with shape $\psi>0$ and scale $\sigma>0$, the expected <Fisher information matrix> for one observation is
$$
I(\psi,\sigma)=\frac1{1+2\psi}\begin{pmatrix}2/(1+\psi)&1/[\sigma(1+\psi)]\\1/[\sigma(1+\psi)]&1/\sigma^2\end{pmatrix}.
$$
The <orthogonalization equation for two statistical parameters> gives $\partial_\psi\sigma=-\sigma/(1+\psi)$ at fixed new nuisance coordinate. Thus $\eta=\sigma(1+\psi)$ is a global <interest-respecting reparametrization>. In coordinates $(\psi,\eta)$, the information is diagonal with entries $(1+\psi)^{-2}$ and $[\eta^2(1+2\psi)]^{-1}$.
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