 Solution

For parameter of interest $\psi$ and nuisance parameter $\lambda$, the profile likelihood has log form ${\ell }_p(\psi )=\ell (\psi ,{\hat{\lambda }}_{\psi })$, where ${\hat{\lambda }}_{\psi }$ is the constrained nuisance maximum. Profiling optimizes over the nuisance dimension but does not account for its local volume or conditioning geometry. The modified profile likelihood corrects this loss.

Use ancillary sample coordinates $(\hat{\psi },\hat{\lambda },a)$, and let $j_{\lambda \lambda }(\psi ,{\hat{\lambda }}_{\psi })=-{\ell }_{\lambda \lambda }$ be the nuisance block of observed information. Holding $\hat{\psi },a$ fixed, put $D(\psi )=|\partial {\hat{\lambda }}_{\psi }/\partial \hat{\lambda }|$. The Barndorff-Nielsen modified profile likelihood is, up to data-only factors,This combines a nuisance-information determinant and a data-coordinate Jacobian determinant. Implicit differentiation of ${\ell }_{\lambda }(\psi ,{\hat{\lambda }}_{\psi })=0$ gives $\partial {\hat{\lambda }}_{\psi }/\partial \hat{\lambda }=j_{\lambda \lambda }^{-1}{\ell }_{\lambda ;\hat{\lambda }}$, so equivalentlyThe semicolon denotes differentiation in the varying sample coordinate, rather than a parameter derivative. This form also explains the positive adjustment in Question 2: in that canonical conditional problem the mixed sample derivative is data-only, leaving the positive half-log curvature. The adjustment sign cannot be copied between unrelated constructions without tracking their Jacobians.

In an appropriate orthogonal nuisance formulation, the Cox-Reid adjusted profile likelihood uses ${\ell }_p-\frac{1}{2}\log |j_{\lambda \lambda }|$. It is a useful simpler adjustment, but discarding $D$ is not universally legitimate. For normal data with interest ${\sigma }^2$ and nuisance $\mu$, the constrained and full nuisance fits both equal $\bar{X}$, so $D=1$ and $j_{\mu \mu }=n/{\sigma }^2$. Thusrecovering the residual degrees of freedom and fitted variance ${\sum }_i(X_i-\bar{X}{)}^2/(n-1)$. These adjustments target nuisance-related bias and higher-order likelihood inference; they are approximations to conditional or marginal constructions, not an assertion that profiling has become an exact full sampling density.