Semiparametric models contain a finite-dimensional target and an infinite-dimensional nuisance law. Their first-order geometry lives in the Hilbert space \(L_0^2(P)\). Scores generate tangent directions, pathwise derivatives identify influence functions, and orthogonal projection produces the unique influence function with minimum variance.
Regular parametric submodels
Definition. Let \(\mathcal P\) be a model containing \(P\). A one-dimensional regular parametric submodel through \(P\) is a path \(\{P_t:|t|\lt\varepsilon\}\subseteq\mathcal P\), \(P_0=P\), that is quadratic-mean differentiable at zero: \[ \int\left[ \sqrt{\mathrm{d}P_t} -\sqrt{\mathrm{d}P} -\frac t2s\sqrt{\mathrm{d}P} \right]^2=o(t^2) \] for a score \(s\in L_0^2(P)\).
Definition (tangent space). The tangent set is the collection of scores of regular submodels through \(P\). Its closed linear span in \(L_0^2(P)\) is the tangent space \(\mathcal T_P\).
The closure matters because information bounds are Hilbert-space projection problems. In a fully nonparametric dominated model, \(\mathcal T_P\) is often all of \(L_0^2(P)\); restrictions such as known propensity scores, moment conditions, or conditional independence reduce it.
Pathwise differentiability
Definition. A scalar functional \(\Psi:\mathcal P\to\mathbb R\) is pathwise differentiable at \(P\) if there exists \(\phi\in L_0^2(P)\) such that, along every regular submodel with score \(s\), \[ \left.\frac{\mathrm{d}}{\mathrm{d}t} \Psi(P_t)\right|_{t=0} =P(\phi s). \] Any such \(\phi\) is an influence function for \(\Psi\) relative to the model.
The derivative is a continuous linear functional on the tangent space. By the Riesz representation theorem, it has a unique representative in \(\mathcal T_P\). That representative is the canonical gradient or efficient influence function.
Canonical-gradient projection. If \(\phi\) is any influence function, then \[ \phi_{\mathrm{eff}}=\Pi_{\mathcal T_P}\phi \] is also an influence function, is unique in \(\mathcal T_P\), and satisfies \[ P\phi_{\mathrm{eff}}^2 \le P\tilde\phi^2 \] for every influence function \(\tilde\phi\).
Proof. For each score \(s\in\mathcal T_P\), \[ P\{(\phi-\Pi_{\mathcal T_P}\phi)s\}=0, \] so the projection represents the same pathwise derivative. If \(\tilde\phi\) is another influence function, then \(P\{(\tilde\phi-\phi_{\mathrm{eff}})s\}=0\) for every \(s\in\mathcal T_P\). Since \(\phi_{\mathrm{eff}}\in\mathcal T_P\), the difference is orthogonal to \(\phi_{\mathrm{eff}}\), and Pythagoras gives \[ P\tilde\phi^2 =P\phi_{\mathrm{eff}}^2 +P(\tilde\phi-\phi_{\mathrm{eff}})^2. \]
Efficiency bound
Semiparametric information bound. Every regular asymptotically linear estimator of \(\Psi(P)\) has an influence function \(\phi\) satisfying the pathwise derivative equation. Therefore its asymptotic variance is at least \[ P\phi_{\mathrm{eff}}^2. \] An estimator with expansion \[ \hat\Psi-\Psi(P) =P_n\phi_{\mathrm{eff}}+o_p(n^{-1/2}) \] is semiparametrically efficient.
Proof. Restrict the estimator to every regular parametric submodel. Le Cam's third lemma and regularity give the covariance identity \(P(\phi s)=\dot\Psi_P(s)\), so \(\phi\) is an influence function. The projection theorem then yields \(P\phi^2\ge P\phi_{\mathrm{eff}}^2\).
Elementary functionals
Example (mean). In the nonparametric model, \[ \Psi(P)=PY. \] Along a submodel with score \(s\), \[ \left.\frac{\mathrm{d}}{\mathrm{d}t}P_tY\right|_0 =P(Ys)=P\{(Y-PY)s\}. \] Hence \[ \phi_{\mathrm{eff}}(Y)=Y-PY. \]
Example (variance). For \[ \Psi(P)=P(Y-\mu)^2,\qquad \mu=PY, \] differentiation gives \[ \phi_{\mathrm{eff}}(Y) =(Y-\mu)^2-\Psi(P). \] The derivative of \(\mu\) disappears because \(P(Y-\mu)=0\).
Average treatment effect
Let \(O=(X,A,Y)\), \(A\in\{0,1\}\), and under consistency, conditional exchangeability, and positivity define \[ \Psi(P) =P\{m_1(X)-m_0(X)\}, \] where \[ m_a(x)=\mathbb E(Y\mid A=a,X=x), \qquad e(x)=P(A=1\mid X=x). \]
Efficient influence function for the ATE. In the nonparametric observed-data model, \[ \begin{aligned} \phi_{\mathrm{ATE}}(O) &= \frac{A}{e(X)}\{Y-m_1(X)\} -\frac{1-A}{1-e(X)}\{Y-m_0(X)\}\\ &\quad{} +m_1(X)-m_0(X)-\Psi(P). \end{aligned} \]
Proof outline. Factor the observed density into \(p(y\mid a,x)p(a\mid x)p(x)\), and decompose every score orthogonally into outcome, treatment, and covariate components. Differentiating \(m_a(x)\) along the outcome component produces \[ \frac{1_{\{A=a\}}}{P(A=a\mid X)} \{Y-m_a(X)\}. \] Differentiating the marginal expectation over \(X\) produces \(m_1(X)-m_0(X)-\Psi(P)\). The treatment-score derivative is zero because the target does not depend on \(p(a\mid x)\). Summing the orthogonal components gives the displayed function, which lies in the full tangent space and is therefore canonical.
One-step estimation and remainder structure
Given nuisance estimates \(\hat\eta\), a one-step estimator is \[ \hat\Psi_{\mathrm{1step}} =\Psi(\hat P)+P_n\phi_{\hat\eta}. \] A useful expansion is \[ \hat\Psi_{\mathrm{1step}}-\Psi(P) =(P_n-P)\phi_{\eta_0} +(P_n-P)(\phi_{\hat\eta}-\phi_{\eta_0}) +R_2(\hat\eta,\eta_0). \] The first term is asymptotically Gaussian. The second is an empirical-process term, and \(R_2\) is a deterministic second-order bias.
Product-rate remainder for the ATE. For the augmented inverse-probability estimator, the leading bias is proportional to \[ P\left[ \frac{\hat e-e}{\hat e} (\hat m_1-m_1) \right] + P\left[ \frac{\hat e-e}{1-\hat e} (\hat m_0-m_0) \right]. \] Under positivity, it is \(o_p(n^{-1/2})\) if the relevant \(L^2(P)\) nuisance-error products are \(o_p(n^{-1/2})\).
Proof. Take conditional expectations of the augmented score given \(X\), substitute the true regression functions and propensity, and cancel the first-order terms. Cauchy-Schwarz bounds each remaining expectation by the product of the two nuisance \(L^2\) errors times a positivity constant.
Orthogonality and cross-fitting
Definition (Neyman orthogonality). A score \(\psi(O;\theta,\eta)\) is orthogonal at \((\theta_0,\eta_0)\) if \[ \left.\frac{\mathrm{d}}{\mathrm{d}t} P\psi(O;\theta_0,\eta_0+t h) \right|_{t=0}=0 \] for every nuisance direction \(h\).
Orthogonality makes nuisance error enter at second order. Cross-fitting estimates \(\eta\) on folds excluding the observation on which \(\psi\) is evaluated. Conditional on the training folds, the evaluation terms behave like independent observations with fixed nuisance estimates, reducing Donsker-class requirements. Cross-fitting does not remove the product-rate remainder, positivity, or moment conditions.
Irregular functionals
If the pathwise derivative is not continuous in the \(L^2(P)\) tangent norm, no square-integrable influence function exists and regular root-\(n\) estimation is impossible. Point evaluation of an unrestricted density is the standard example: increasingly concentrated score perturbations can change \(f(x_0)\) without bounded \(L^2\) derivative. Smoothing or stronger model restrictions then determine a slower rate.
Empirical-process conditions used to control estimated influence functions are developed in empirical processes; the Hilbert-space projection foundations are in functional analysis.