 Solution

Partition a regular parameter as $(\psi ,\lambda )$, with interest $\psi$ and nuisance $\lambda$. Orthogonal statistical parameters have zero cross-block of expected Fisher information,This may hold at one parameter value or throughout a parametrization. It is expected-information orthogonality, and does not generally force the observed information cross-block to vanish in each sample.

First, the asymptotic covariance of the joint maximum-likelihood estimator is the inverse information. Orthogonality makes this matrix block diagonal, giving first-order asymptotic independence of interest and nuisance fits. The efficient information with unknown nuisance isFor orthogonal parameters it equals $I_{\psi \psi }$. Thus estimating the nuisance creates no first-order information loss relative to knowing it, at the orthogonal point. This is a local asymptotic statement, not a general finite-sample independence theorem.

Second, orthogonality stabilizes constrained nuisance estimates. Differentiating ${\ell }_{\lambda }(\psi ,{\tilde{\lambda }}_{\psi })=0$ givesWhen the expected cross-information vanishes, regular sampling fluctuations give $j_{\lambda \psi }=O_p(\sqrt{n})$ and $j_{\lambda \lambda }=O_p(n)$. The derivative is therefore $O_p(n^{-1/2})$ locally. Changing the interest parameter by $O_p(n^{-1/2})$ changes its constrained nuisance fit by only $O_p(n^{-1})$, smaller than the ordinary nuisance estimation error. This reduces sensitivity to how the nuisance is fitted and simplifies higher-order likelihood adjustments.

For scalar interest, local orthogonal nuisance reparametrization can often be constructed explicitly. Keep $\psi$ and write the original nuisance as $\lambda (\psi ,\eta )$. The transformed interest score is $U_{\psi }+{\lambda }_{\psi }^{\mathsf{T}}{U}_{\lambda }$. Its covariance with the new nuisance score vanishes whenWith smooth nonsingular nuisance information, this ordinary differential equation has a local solution for suitable initial nuisance coordinates. For several interest coordinates the corresponding equations need compatibility conditions; global orthogonality is not automatic.

The normal model gives a transparent example. For independent observations with mean $\mu$ and variance $\nu$, direct score expectations giveThe cross-information is zero because the centered normal third moment is zero. When fitting $\nu$ at fixed $\mu$,so the change is indeed quadratic in a local change of the mean.

Orthogonality is also useful for removing nuisance effects beyond first order. Take variance $\nu$ as interest and mean $\mu$ as nuisance. Put $Q={\sum }_i(X_i-\overline{X}{)}^2$. The profile log-likelihood is ${\ell }_p(\nu )=-\frac{n}{2}\log \nu -Q/(2\nu )$ up to a constant. The constrained mean is ${\tilde{\mu }}_{\nu }=\overline{X}$ for all $\nu$, so its sample-coordinate Jacobian determinant is one, and $j_{\mu \mu }=n/\nu$. The information adjustment givesThis agrees with the exact likelihood based on $Q/\nu \sim {\chi }_{n-1}^2$: estimating the mean has used one degree of freedom. In this setting the Cox-Reid adjusted profile likelihood has the same form. The exact distribution follows by projecting the centered normal vector onto the $(n-1)$-dimensional subspace orthogonal to the constant vector; an orthonormal basis there supplies $n-1$ independent standard normal coordinates whose squares sum to $Q/\nu$.

The example shows both the benefit and the limits. Orthogonality organizes first-order precision and nuisance stability; it does not eliminate all higher-order nuisance effects. Nor does it identify an ancillary statistic or justify dropping the sample-coordinate factor in every modified profile likelihood. In the exponential regression calculation that factor cancels the information adjustment, so an information-only correction would give a different answer.