Interest-respecting reparametrization

An interest-respecting reparametrization preserves the parameter of interest, or changes it only by a one-to-one function of itself. The nuisance parameter may depend on both old parameters, provided the complete transformation is smoothly invertible. Thus the level sets representing the inferential target are unchanged. Orthogonality can be sought by choosing the new nuisance coordinate along solutions of the orthogonalization equation for two statistical parameters.

Write the old nuisance parameter as $\lambda =g(\psi ,\eta )$ with $g_{\eta }\ne 0$. The transformed score functions are $u_{\psi }^{\mathrm{new}}=u_{\psi }+g_{\psi }{u}_{\lambda }$ and $u_{\eta }=g_{\eta }{u}_{\lambda }$. Their expected cross product is $g_{\eta }(I_{\psi \lambda }+g_{\psi }{I}_{\lambda \lambda })$. Setting it to zero gives the displayed ordinary differential equation along fixed-$\eta$ curves, assuming positive nuisance Fisher information. Smooth solutions with distinct initial conditions give a local interest-respecting orthogonal coordinate system.