= Modified profile likelihood for inverse Gaussian shape
{title2=$\ell_m(\psi)=\tfrac{n-1}{2}\log\psi-\tfrac\psi2\left(\sum_iY_i^{-1}-n/\bar Y\right)+\text{constant}$}
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 $(n-1)/(\sum_iY_i^{-1}-n/\bar Y)$.
Back to article page
