Modified profile likelihood for inverse Gaussian shape (${\ell }_m(\psi )=\frac{n-1}{2}\log \psi -\frac{\psi }{2}\left({\sum }_i{Y}_i^{-1}-n/\bar{Y}\right)+\text{constant}$)

For independent Inverse Gaussian distributions with shape $\psi$ and mean $\lambda$, the constrained and full nuisance maximum-likelihood estimates both equal $\bar{Y}$, giving a unit sample-coordinate Jacobian determinant. The nuisance observed information there is $n\psi /{\bar{Y}}^3$. Hence subtracting half its logarithm from the profile log-likelihood replaces the coefficient $n/2$ of $\log \psi$ by $(n-1)/2$. For a nonconstant sample the modified fit is $(n-1)/({\sum }_i{Y}_i^{-1}-n/\bar{Y})$.