Cox-Reid adjusted profile likelihood (${\ell }_{\mathrm{CR}}(\psi )={\ell }_p(\psi )-\frac{1}{2}\log |j_{\lambda \lambda }(\psi ,{\hat{\lambda }}_{\psi })|$)

For a regular interest/nuisance parameterization with parameter orthogonality, this nuisance-information adjustment approximates a conditional likelihood construction and reduces first-order nuisance effects. It is related to modified profile likelihood but is not a universal substitute for its sample-coordinate Jacobian. When the constrained and full nuisance fits coincide in the relevant ancillary coordinates, that Jacobian is one and the two expressions agree. For normal variance with unknown mean, it replaces the profile's $n$ by the residual $n-1$ in the logarithmic variance term.