= Inverse Gaussian shape estimation and nuisance orthogonality
{title2=$I_{\psi\lambda}=0,\quad I_{\psi\psi}=1/(2\psi^2)$}
For the <Inverse Gaussian distribution> parametrized by shape $\psi$ and mean $\lambda$, the joint <maximum-likelihood estimators> are $\widehat\lambda=\bar Y$ and $\widehat\psi=(\overline{Y^{-1}}-1/\bar Y)^{-1}$. With known $\lambda$, the shape estimator is instead $(\overline{Y^{-1}}-2/\lambda+\bar Y/\lambda^2)^{-1}$. <Parameter orthogonality> gives the same <Wald test> formula $n(\widehat\psi-\psi_0)^2/(2\widehat\psi^2)$ in both fits. The two estimators differ by $O_p(n^{-1})$ under regular model sampling, so the test statistics agree to first order under the null; they need not be numerically equal in finite samples.
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