The Cox model contains a finite-dimensional target \(\beta\) and an infinite-dimensional nuisance \(\Lambda_0\), together with nuisance laws for covariates and censoring. Semiparametric efficiency asks which component of the ordinary score remains after every nuisance direction has been projected away.
Observed-data model and tangent directions
Definition. For one subject, write \[ \lambda_{\beta,\Lambda}(t\mid\mathcal F_{t-}) =Y(t)e^{\beta^\top Z(t)}\,\mathrm{d}\Lambda(t). \] A regular parametric submodel through \((\beta_0,\Lambda_0)\) may be written \[ \beta_\varepsilon=\beta_0+\varepsilon a, \qquad \mathrm{d}\Lambda_\varepsilon(t) =\{1+\varepsilon h(t)\}\,\mathrm{d}\Lambda_0(t), \] where \(a\in\mathbb R^p\) and \(h\in L^2(\nu)\) for the event-information measure \[ \nu(\mathrm{d}t)=s^{(0)}(\beta_0,t)\,\mathrm{d}\Lambda_0(t). \]
Proposition (pathwise scores). Along these paths, the regression and baseline-hazard scores are \[ \dot\ell_\beta[a] =a^\top\int_0^\tau Z(t)\,\mathrm{d}M(t), \qquad \dot\ell_\Lambda[h] =\int_0^\tau h(t)\,\mathrm{d}M(t), \] where \[ M(t)=N(t)-\int_0^tY(u)e^{\beta_0^\top Z(u)} \,\mathrm{d}\Lambda_0(u). \]
Proof. Differentiate the point-process log-likelihood \[ \int\{\log\mathrm{d}\Lambda+\beta^\top Z\}\,\mathrm{d}N -\int Ye^{\beta^\top Z}\,\mathrm{d}\Lambda. \] For \(\beta\), the derivative is \[ a^\top\int Z\{\mathrm{d}N -Ye^{\beta_0^\top Z}\mathrm{d}\Lambda_0\} =a^\top\int Z\,\mathrm{d}M. \] For the multiplicative baseline path, the derivative is \[ \int h\,\mathrm{d}N-\int hYe^{\beta_0^\top Z}\,\mathrm{d}\Lambda_0 =\int h\,\mathrm{d}M. \]
Nuisance tangent space
Definition. The baseline-hazard tangent space is the closure in \(L_0^2(P)\) of \[ \mathcal T_\Lambda =\left\{ \int_0^\tau h(t)\,\mathrm{d}M(t): h\in L^2(\nu) \right\}. \] Additional tangent spaces arise from the covariate and censoring laws. Under noninformative censoring and unrestricted covariate distribution, their scores are orthogonal to event martingale integrals after conditioning on predictable history.
Lemma (martingale inner product). For predictable square-integrable \(f,g\), \[ E\left[ \left(\int f\,\mathrm{d}M\right) \left(\int g\,\mathrm{d}M\right) \right] =E\int_0^\tau f(t)g(t)Y(t)e^{\beta_0^\top Z(t)} \,\mathrm{d}\Lambda_0(t). \]
Proof. This is the martingale isometry, because \[ \mathrm{d}\langle M\rangle(t) =Y(t)e^{\beta_0^\top Z(t)} \,\mathrm{d}\Lambda_0(t). \]
Projection onto the nuisance complement
Theorem (efficient score). Let \[ e(t)= \frac{E\{Y(t)e^{\beta_0^\top Z(t)}Z(t)\}} {E\{Y(t)e^{\beta_0^\top Z(t)}\}}. \] The efficient score for \(\beta\) is \[ \widetilde\ell_\beta =\int_0^\tau\{Z(t)-e(t)\}\,\mathrm{d}M(t). \] Its covariance, the efficient information, is \[ \widetilde I =E\int_0^\tau \{Z(t)-e(t)\}^{\otimes2} Y(t)e^{\beta_0^\top Z(t)} \,\mathrm{d}\Lambda_0(t) =\Sigma. \]
Proof. Project the vector integrand \(Z(t)\) onto deterministic functions of time in the weighted Hilbert space with inner product \[ \langle f,g\rangle =E\int f(t)^\top g(t) Y(t)e^{\beta_0^\top Z(t)} \,\mathrm{d}\Lambda_0(t). \] At each \(t\), the weighted least-squares minimizer of \(E[Y e^{\beta_0^\top Z}\|Z-h(t)\|^2]\) is \(h(t)=e(t)\). Thus \(Z-e\) is orthogonal to every baseline direction: \[ E\int\{Z(t)-e(t)\}h(t) Y(t)e^{\beta_0^\top Z(t)} \,\mathrm{d}\Lambda_0(t)=0. \] Subtracting the projection from the raw score gives the efficient score. Its covariance follows from the martingale isometry.
Geometry. Risk-set centering is exactly an orthogonal projection. Partial likelihood did not discover an accidental algebraic simplification: its score is the component of the full regression score orthogonal to the baseline-hazard tangent space.
Efficient influence function and lower bound
Theorem. If \(\widetilde I\) is nonsingular, the efficient influence function for \(\beta\) is \[ \widetilde\phi_\beta =\widetilde I^{-1} \int_0^\tau\{Z(t)-e(t)\}\,\mathrm{d}M(t). \] Every regular asymptotically linear estimator \[ \sqrt n(T_n-\beta_0) =n^{-1/2}\sum_{i=1}^n\phi(O_i)+o_P(1) \] satisfies \[ \operatorname{Var}\{\phi(O)\} \succeq \widetilde I^{-1}. \] Equality holds exactly when \(\phi=\widetilde\phi_\beta\) in \(L^2(P)\).
Proof. A valid influence function must reproduce the pathwise derivative in every score direction: \[ E(\phi\dot\ell_\beta^\top)=I_p, \qquad E(\phi g)=0\quad(g\in\mathcal T_{\mathrm{nuis}}). \] The efficient score lies in the nuisance orthogonal complement and has covariance \(\widetilde I\), so \(\widetilde I^{-1}\widetilde\ell_\beta\) satisfies these equations. Any other influence function is \(\widetilde\phi_\beta+r\), where \(r\) is orthogonal to the efficient score. Pythagoras gives \[ \operatorname{Var}(\phi) =\widetilde I^{-1}+\operatorname{Var}(r) \succeq\widetilde I^{-1}. \]
Efficiency of partial likelihood
Corollary. The Cox partial-likelihood estimator is semiparametrically efficient for \(\beta\) under the correctly specified proportional-hazards model and noninformative censoring.
Proof. The large-sample expansion from page 6 is \[ \sqrt n(\widehat\beta-\beta_0) =\widetilde I^{-1} \frac1{\sqrt n}\sum_{i=1}^n \int\{Z_i(t)-e(t)\}\,\mathrm{d}M_i(t) +o_P(1). \] Its influence function equals \(\widetilde\phi_\beta\), so its asymptotic covariance attains the lower bound.
Functionals involving the baseline hazard
For a smooth target \[ \psi(\beta,\Lambda_0), \] the efficient influence function combines the regression and baseline components. For example, at fixed \(z,t\), \[ \psi=S(t\mid z) =\exp\{-e^{\beta^\top z}\Lambda_0(t)\}, \] so the chain rule gives \[ \widetilde\phi_{S_z,t} =-S(t\mid z)e^{\beta_0^\top z} \left[ \widetilde\phi_{\Lambda,t} +\Lambda_0(t)z^\top\widetilde\phi_\beta \right]. \] The baseline influence component must itself be orthogonalized against the regression score. Plugging in a Nelson-Aalen variance while treating \(\widehat\beta\) as fixed omits this covariance term.
When efficiency calculations change
Remark. The tangent space changes under:
- dependent censoring modeled through inverse-probability weights;
- case-cohort or nested case-control sampling;
- unknown treatment or observation mechanisms;
- time-varying coefficients, frailty, or informative cluster size;
- high-dimensional nuisance functions estimated by machine learning.
In each case the correct procedure is to specify the observed-data law, derive all nuisance scores, project the target score, and only then write an influence function. Reusing the ordinary Cox information matrix outside its model is not an efficiency argument.
The general Hilbert-space construction, pathwise differentiability, and one-step estimation are developed in semiparametric inference.