Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 41 / 6 / v / Solution

Partition a regular parameter into interest $\psi$ and nuisance $\lambda$. Parameter orthogonality means the cross-block of the expected Fisher information matrix vanishes:It is expected-information orthogonality, not a demand that every observed cross derivative be zero. To first order, the asymptotic normal distribution of the joint maximum-likelihood estimator has independent interest and nuisance blocks. Its efficient interest information simplifies from the Schur complement $I_{\psi \psi }-I_{\psi \lambda }{I}_{\lambda \lambda }^{-1}{I}_{\lambda \psi }$ to $I_{\psi \psi }$.

For scalar interest, new nuisance coordinates $\eta$ can locally be chosen by the local orthogonal nuisance reparametrizationIndeed the new interest score is $U_{\psi }+({\partial }_{\psi }\lambda {)}^{\top }{U}_{\lambda }$, and its expected product with the transformed nuisance score is zero by this equation. Smoothness, nonsingular nuisance information and suitable initial coordinates provide the local ordinary-differential-equation construction. Simultaneous orthogonalization for several interest coordinates may face integrability constraints.

For normal location and variance $(\mu ,{\sigma }^2)$, the cross information is zero: the location score is proportional to the centered observation and the variance score to its squared value minus its expectation, whose product has zero expectation by normal symmetry. This is a useful orthogonal parameterization. In general orthogonality reduces nuisance sensitivity and motivates adjusted profile constructions, but it implies neither exact estimator independence nor an exact nuisance-free conditional likelihood.