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If the parametric score function has zero conditional expectation given a covariate $X$, it is orthogonal to the centered functions of $X$ that form the covariate-density nuisance tangent space. Its efficient score then equals its parametric score function, and the unknown covariate distribution causes no loss of Fisher information. This applies to Gaussian regression scores and more generally to regular conditional models with unrestricted covariate distribution and no additional nuisance components.

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.

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.

Fix $P=P_{\theta ,\eta }$. Let $\mathcal{N}$ be the nuisance tangent space: the closed linear span in $L^2(P)$ of score functions of statistical paths that vary only the nuisance parameter $\eta$. By the preceding argument, $\mathcal{N}$ is contained in the mean-zero L2 space $L_0^2(P)$.

Let ${\Pi }_{\mathcal{N}}$ denote orthogonal projection onto this closed subspace of a Hilbert space. The efficient score and scalar efficient information areThe efficient score is the component of the parametric score function that cannot be reproduced by changing the nuisance parameter. The efficient information is its squared L2 norm; it can be zero, so positivity must not be assumed in the definition.

The map $h\mapsto Ph$ is a bounded linear functional on L2 space, since $|Ph|\le \Vert h{\Vert }_2$ by the Cauchy-Schwarz inequality. Its kernel, the mean-zero L2 space, is therefore closed. Every nuisance score function and the parametric score function $\dot{\ell }$ are centered, so both ${\Pi }_{\mathcal{N}}\dot{\ell }$ and $\tilde{\ell }$ are centered. HenceMoreover, the efficient score belongs to the orthogonal complement of the nuisance tangent space. Writing $\dot{\ell }=\tilde{\ell }+{\Pi }_{\mathcal{N}}\dot{\ell }$ givesThis is the efficient-score projection identity; it remains valid when the efficient information is zero.

Vary only the probability density function of $X$, using ${\eta }_t=\eta (1+tb)$ with bounded $b$ and $E_{\eta }b(X)=0$. The nuisance score function is $b(X)$, giving the statistical tangent setIts closed linear span is the nuisance tangent space of all centered $L^2(\eta )$ functions of $X$, by density of bounded centered scores.

The conditional expectation of the parametric score function given $X$ is zero:It is therefore orthogonal to this nuisance tangent space. Its orthogonal projection onto that space vanishes, so the efficient score is unchanged. By independence and $E{\epsilon }^2={\sigma }^2$,These equal the parametric score function and Fisher information when $\eta$ is known. There is no loss of information from the unknown covariate density. This is adaptivity to an unknown covariate distribution; it follows from score orthogonality, without having to estimate the nuisance parameter.

First use the additional admitted form of the efficient score. Write $\tilde{\ell }=e\zeta (X)$ and let ${\tau }^2=E_f{\epsilon }^2>0$. The difference $\dot{\ell }-\tilde{\ell }$ is the orthogonal projection onto the nuisance tangent space. Part (c) makes every $e\psi (X)$ orthogonal to that space. Therefore, for all $\psi \in L^2(v)$,The function in braces belongs to L2 space; choosing it as $\psi$ shows it vanishes $v$-almost everywhere. Hence the required conditional deduction isUnder the regular tail condition $sf(s)\to 0$ at both infinities, integration by parts gives $\int s{f}^{\prime }(s)ds=-1$, and the formula becomes $e{h}_{\theta }(x)/{\tau }^2$. This also confirms the sign.

The admitted product form is an extra restriction; it does not follow for every independent-error regression model. To locate the restriction precisely, suppose the error-density nuisance statistical paths preserve both normalization and mean to first order. Their closed mean-preserving error tangent space is $\{\gamma :E_f\gamma =E_f(\epsilon \gamma )=0\}$. The full nuisance tangent space is the orthogonal sum of this space and the centered functions of $X$.

For completeness, bounded functions satisfying the two constraints are dense in the error space. Truncate an arbitrary element, subtract its expected value, and then subtract a multiple of a fixed bounded centered function $q$ with $E_f(\epsilon q)\ne 0$. Such a $q$ exists by truncating $\epsilon$, since ${\tau }^2>0$. The correction coefficients tend to zero by the Cauchy-Schwarz inequality, so the corrected truncations converge in L2 space.

Let $m=E_v{h}_{\theta }(X)$. With $E_f\rho =0$ and $E_f(\epsilon \rho )=1$, the orthogonal projection of $h_{\theta }(X)\rho (\epsilon )$ onto the error nuisance space is $m\{\rho (\epsilon )-\epsilon /{\tau }^2\}$; its projection onto the covariate nuisance space is zero. Thus the general efficient score in independent-error regression isThe orthogonality of the two terms gives the displayed efficient information. For a normal distribution of the error, $\rho (e)=e/{\tau }^2$, so this reduces to the admitted formula. It also does so when $h_{\theta }$ is constant.

A concrete counterexample to the generality of the admission is $g_{\theta }(x)=\theta x$, with $X$ uniform on $[-1,1]$ and independent standard logistic distribution error. Here $m=0$ and $\rho (e)=\tanh (e/2)$, so the actual efficient score is $x\tanh (e/2)$. It cannot equal $e\zeta (x)$ because $\tanh (e/2)/e$ is not constant. This score function is already orthogonal to every nuisance score function: its factor $X$ is centered against error-only directions, and its factor $\rho (\epsilon )$ is centered against covariate-only directions. The requested formula is valid under its stated additional admission, with the general independent-error formula above explaining its limits.