Efficient score ($\tilde{\ell }=\dot{\ell }-{\Pi }_{\mathcal{N}}\dot{\ell }$)

The efficient score is the residual after orthogonal projection of the parametric score function onto the nuisance tangent space. It is centered and orthogonal to every nuisance direction. Its squared L2 norm is the efficient information.

With unknown independent covariate and centered error distributions, take the regular mean-preserving error tangent space, let $h={\partial }_{\theta }{g}_{\theta }$, $\rho =-f^{\prime }/f$ and ${\tau }^2=E_f{\epsilon }^2>0$. Under $E_f\rho =0$, $E_f(\epsilon \rho )=1$ and finite second moments, orthogonal projection removes $(E_{v}h)(\rho -\epsilon /{\tau }^2)$ from $h\rho$. Thus the efficient score and efficient information are $\tilde{\ell }=(h-E_{v}h)\rho +(E_{v}h)\epsilon /{\tau }^2$ and $\tilde{I}={\mathrm{Var}}_v(h)E_f{\rho }^2+(E_{v}h{)}^2/{\tau }^2$. The reduction to $h\epsilon /{\tau }^2$ is valid for a normal distribution of errors or for constant $h$, but need not hold otherwise. Centered covariates and logistic distribution errors give the counterexample $\tilde{\ell }=X\tanh (\epsilon /2)$.

The decomposition $\dot{\ell }={\Pi }_{\mathcal{N}}\dot{\ell }+\tilde{\ell }$ is orthogonal. Taking the inner product with the efficient score gives $P(\dot{\ell }\tilde{\ell })=\tilde{I}$. Centering of the efficient score follows because the nuisance tangent space and the parametric score function lie in the closed mean-zero L2 space.

The efficient information for a scalar target statistical parameter is the squared L2 norm of its efficient score. It cannot exceed the parametric Fisher information, by the Pythagorean theorem in an inner-product space. It can vanish when first-order target variation can be reproduced by nuisance variation.