 Solution

Parameters $\psi$ and $\lambda$ are orthogonal parameters if the cross block of the expected Fisher information matrix is zero throughout the relevant parameter region:This is a statement about expected information, not a requirement that every observed cross derivative vanish. For regular maximum-likelihood estimators, $\sqrt{n}(\hat{\theta }-\theta )$ has limiting covariance equal to the inverse per-observation Fisher information matrix. Block diagonality therefore gives asymptotically independent fitted interest and nuisance parameters. The efficient information $I_{\psi \psi }-I_{\psi \lambda }{I}_{\lambda \lambda }^{-1}{I}_{\lambda \psi }$ reduces to $I_{\psi \psi }$ at an orthogonal parametrization, although nuisance estimation can still matter at higher orders.

For scalar interest and nuisance, retain $\psi$ and write the old nuisance as $\lambda =\lambda (\psi ,\eta )$. The transformed cross information is ${\lambda }_{\eta }(I_{\psi \lambda }+{\lambda }_{\psi }{I}_{\lambda \lambda })$. Hence an orthogonal nuisance coordinate can locally be found fromsubject to local solvability and a nonsingular coordinate change. Global coordinates need not follow from this local construction. For the normal distribution with separately varying mean and variance parameters $(\mu ,v)$, the scores are $(Y-\mu )/v$ and $-1/(2v)+(Y-\mu {)}^2/(2v^2)$; their product has zero expectation by symmetry. Thus the per-observation information is diagonal with entries $1/v$ and $1/(2v^2)$.