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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.

For bounded measurable $g$ with $Pg=0$, the formula $p_t=p(1+tg)$ defines a normalized positive probability density function for $|t|\Vert g{\Vert }_{\infty }<1$. A uniform Taylor expansion of the square root proves differentiability in quadratic mean with score function $g$: the squared remainder is $O(t^{4}P{g}^4)$. Such paths realize all bounded centered directions of an unrestricted density model.

For a regular scalar model with score function $S_{\theta }$ and Fisher information $I(\theta )={\mathbb{E}}_{\theta }{S}_{\theta }^2$, an estimator $T$ with mean $\psi (\theta )$ satisfiesIndeed, differentiating ${\mathbb{E}}_{\theta }T=\psi (\theta )$ gives ${\mathrm{Cov}}_{\theta }(T,S_{\theta })={\psi }^{\prime }(\theta )$, and the Cauchy-Schwarz inequality gives the bound.

A dominated statistical path is differentiable in quadratic mean at $P$ if its square-root probability density function has an L2 space derivative $(1/2)g\sqrt{p}$, with $g\in L^2(P)$. The score function is $g$. This formulation controls mass near zeros of $p$ as well as the derivative on the support of $P$; a pointwise log-density derivative is insufficient by itself.

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.

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.

For $X\sim N_p(\theta ,\Sigma )$ with known nonsingular covariance matrix, the score function is ${\Sigma }^{-1}(X-\theta )$ and the per-observation Fisher information matrix is ${\Sigma }^{-1}$. In particular, covariance $I_p$ gives information $I_p$.

At an interior maximum-likelihood fit of a logistic-regression model containing an intercept, the intercept score function isConsequently the sum of the fitted probabilities equals the number of observed successes exactly.

For a regression function $g_{\theta }$, let $h_{\theta }={\partial }_{\theta }{g}_{\theta }$. With independent centered normal distribution error of known variance ${\sigma }^2$, the parametric score function is $h_{\theta }(X)\epsilon /{\sigma }^2$. It has zero conditional expectation given the covariate, so unknown covariate-density nuisance directions do not remove any of its Fisher information.

An influence-function representer is a centered square-integrable function representing derivatives of a statistical functional along all admissible score functions. Representers may differ by a function orthogonal to the statistical tangent space. This pathwise definition is distinct from defining an influence function solely by point-mass contamination paths.

For a continuously differentiable vector-valued function whose domain contains the segment from $x$ to $x+h$, apply the fundamental theorem of calculus componentwise to $t\mapsto F(x+th)$. The resulting integral matrix is valid even when there is no common intermediate point giving a vector mean-value formula. For a score function, it integrates the Hessian matrix along the segment to a candidate estimate.

The location score is the score function for translating a probability density function $f$. The minus sign comes from differentiating $f(y-\theta )$ in $\theta$. Under regular tail conditions, integration by parts gives $E_f\rho =0$ and $E_f(\epsilon \rho )=1$. These identities determine projections of the score function onto the span of the centered error.

For independent-error regression with a zero-mean error, bounded density paths $f_t=f(1+t\gamma )$ must preserve both normalization and the first moment. Their score functions therefore satisfy two constraints. When $0<E_f{\epsilon }^2<\infty$, truncation followed by two small bounded moment corrections shows these scores are dense in the displayed closed subspace of a Hilbert space. By independence, they are orthogonal to every $\epsilon \psi (X)$ with $\psi \in L^2(v)$. The same constraints apply to other paths only under regularity permitting differentiation of the first moment.

The mean-zero L2 space is the kernel of the bounded linear functional $h\mapsto Ph$ on L2 space for a probability measure $P$. It is a closed subspace of a Hilbert space, with inner product $P(hk)$, and is the natural ambient space for score functions and canonical gradients.

For a differentiable-in-quadratic-mean path, normalization gives $⟨(\sqrt{p_t}-\sqrt{p})/t,\sqrt{p_t}+\sqrt{p}⟩=0$. The two factors converge in L2 space to $g\sqrt{p}/2$ and $2\sqrt{p}$. Continuity of the inner product therefore gives $Pg=0$. This proves score function centering without differentiating the density integral.

The nuisance tangent space is the closed linear span of score functions from statistical paths that vary the nuisance parameter while fixing the target statistical parameter. It is a closed subspace of a Hilbert space inside $L_0^2(P)$. Removing its component from a parametric score function leaves the information that nuisance variation cannot imitate.