The asymptotic theory of the Cox estimator combines three ingredients: uniform laws of large numbers for random risk-set sums, a martingale central limit theorem for the score, and a local Taylor expansion of a concave criterion. Writing the assumptions explicitly reveals where censoring, support, and covariate regularity enter.
Population risk-set moments
Definition. For \(k=0,1,2\), define normalized empirical moments \[ \mathbb S_n^{(k)}(\beta,t) =\frac1n\sum_{i=1}^n Y_i(t)e^{\beta^\top Z_i(t)}Z_i(t)^{\otimes k}. \] Their population counterparts are \[ s^{(k)}(\beta,t) =E\{Y(t)e^{\beta^\top Z(t)}Z(t)^{\otimes k}\}, \] with \[ e(\beta,t)=\frac{s^{(1)}(\beta,t)}{s^{(0)}(\beta,t)},\qquad v(\beta,t)=\frac{s^{(2)}(\beta,t)}{s^{(0)}(\beta,t)} -e(\beta,t)e(\beta,t)^\top. \]
Regularity conditions. Fix a compact parameter neighborhood \(B\) of the true \(\beta_0\) and a finite horizon \(\tau\). Assume:
- the subjects are independent and identically distributed event histories;
- \(Z(t)\) is predictable and uniformly bounded on \([0,\tau]\), or has an exponential envelope strong enough for all displayed risk-set moments;
- \(\inf_{(\beta,t)\in B\times[0,\tau]}s^{(0)}(\beta,t)>0\);
- \(\mathbb S_n^{(k)}\to s^{(k)}\) uniformly in probability for \(k=0,1,2\);
- the baseline cumulative hazard satisfies \(\Lambda_0(\tau)<\infty\);
- the information matrix \[ \Sigma =\int_0^\tau v(\beta_0,t) s^{(0)}(\beta_0,t)\,\mathrm{d}\Lambda_0(t) \] is positive definite.
Bounded covariates are convenient rather than essential. Uniform convergence can instead be proved from Glivenko-Cantelli conditions for the indexed at-risk class; the relevant machinery is developed in empirical processes.
Deterministic limit of partial likelihood
Proposition. Up to a term independent of \(\beta\), \[ \frac1n\{\ell_p(\beta)-\ell_p(\beta_0)\} \xrightarrow{P}Q(\beta) \] uniformly on \(B\), where \[ Q(\beta) =\int_0^\tau \left[ (\beta-\beta_0)^\top s^{(1)}(\beta_0,t) -s^{(0)}(\beta_0,t) \log\frac{s^{(0)}(\beta,t)}{s^{(0)}(\beta_0,t)} \right]\mathrm{d}\Lambda_0(t). \] Moreover, \(Q(\beta)\le Q(\beta_0)=0\), with equality only at \(\beta_0\) under identifiability.
Proof. Replace the average event measure by its compensator: \[ \frac1n\sum_i\mathrm{d}N_i(t) =\mathbb S_n^{(0)}(\beta_0,t)\,\mathrm{d}\Lambda_0(t) +o_P(1) \] uniformly after integration against bounded predictable functions. The uniform convergence of risk-set sums then gives \(Q\). For each \(t\), introduce risk-set exponential-tilt weights under \(\beta_0\). Jensen's inequality gives \[ \log E_{\beta_0,t}e^{(\beta-\beta_0)^\top Z} \ge(\beta-\beta_0)^\top E_{\beta_0,t}Z, \] so the integrand in \(Q\) is nonpositive. Strictness follows if no nonzero direction is constant on almost every informative risk set.
Consistency
Theorem. Under the regularity conditions, any maximizer \(\widehat\beta\) of \(\ell_p\) over a compact set containing \(\beta_0\) satisfies \[ \widehat\beta\xrightarrow{P}\beta_0. \]
Proof. For every neighborhood \(U\) of \(\beta_0\), compactness and the unique maximum of \(Q\) give \[ \sup_{\beta\notin U}Q(\beta)<Q(\beta_0). \] Uniform convergence transfers this separation to \(n^{-1}\ell_p\) with probability tending to one. On that event no maximizer can lie outside \(U\). Since \(U\) is arbitrary, consistency follows.
Score central limit theorem
Theorem. Under the correctly specified Cox model, \[ n^{-1/2}U_n(\beta_0)\Rightarrow N_p(0,\Sigma). \]
Proof. The martingale representation is \[ n^{-1/2}U_n(\beta_0) =n^{-1/2}\sum_i\int_0^\tau \{Z_i(t)-\bar Z_n(\beta_0,t)\}\,\mathrm{d}M_i(t). \] Its predictable bracket equals \[ \int_0^\tau V_n(\beta_0,t)\mathbb S_n^{(0)}(\beta_0,t) \,\mathrm{d}\Lambda_0(t), \] which converges in probability to \(\Sigma\). Bounded covariates and positive risk sets make the maximal jump \(O_P(n^{-1/2})\), verifying Lindeberg's condition. The multivariate martingale CLT gives the result.
Asymptotic linearity and normality
Theorem. Under the regularity conditions, \[ \sqrt n(\widehat\beta-\beta_0) =\Sigma^{-1}\frac{1}{\sqrt n}U_n(\beta_0)+o_P(1) \Rightarrow N_p(0,\Sigma^{-1}). \] Equivalently, \[ \widehat\beta-\beta_0 =\frac1n\sum_{i=1}^n\phi_\beta(O_i)+o_P(n^{-1/2}), \] where one version of the influence function is \[ \phi_\beta(O_i) =\Sigma^{-1}\int_0^\tau \{Z_i(t)-e(\beta_0,t)\}\,\mathrm{d}M_i(t). \]
Proof. The score equation and the mean-value theorem give \[ 0=U_n(\widehat\beta) =U_n(\beta_0)-I_n(\widetilde\beta) (\widehat\beta-\beta_0) \] for an intermediate \(\widetilde\beta\). Consistency and uniform convergence imply \(n^{-1}I_n(\widetilde\beta)\xrightarrow{P}\Sigma\). Invert this matrix and combine with the score CLT. Replacing the empirical risk-set mean by its limit in the martingale sum costs \(o_P(1)\), yielding the influence representation.
Corollary. The inverse observed information \[ I_n(\widehat\beta)^{-1} \] consistently estimates the covariance of \(\widehat\beta\). For any fixed \(a\), \[ \frac{a^\top(\widehat\beta-\beta_0)} {\{a^\top I_n(\widehat\beta)^{-1}a\}^{1/2}} \Rightarrow N(0,1). \]
Local alternatives
Proposition. Under local alternatives \(\beta_n=\beta_0+h/\sqrt n\), \[ n^{-1/2}U_n(\beta_0)\Rightarrow N_p(\Sigma h,\Sigma), \qquad \sqrt n(\widehat\beta-\beta_0)\Rightarrow N_p(h,\Sigma^{-1}). \]
Proof. The partial log-likelihood is locally asymptotically quadratic: \[ \ell_p(\beta_0+h/\sqrt n)-\ell_p(\beta_0) =h^\top n^{-1/2}U_n(\beta_0) -\frac12h^\top\Sigma h+o_P(1). \] Le Cam's third lemma shifts the score mean from zero to \(\Sigma h\). The estimator expansion then gives the second limit. LAN and contiguity are treated systematically in statistical decision theory.
Misspecification and sandwich covariance
Theorem (pseudo-true Cox parameter). Suppose the true intensity need not be proportional hazards, but the empirical score converges uniformly to a deterministic map \(\Psi(\beta)\) with unique zero \(\beta^\star\). Under differentiability and nonsingularity of \[ A=-\dot\Psi(\beta^\star), \] the partial-likelihood estimator satisfies \[ \sqrt n(\widehat\beta-\beta^\star) \Rightarrow N_p(0,A^{-1}BA^{-\top}), \] where \(B\) is the limiting covariance of \(n^{-1/2}U_n(\beta^\star)\).
Proof. Uniform convergence and identification imply \(\widehat\beta\to\beta^\star\). Taylor expansion of the score gives \[ \sqrt n(\widehat\beta-\beta^\star) =A^{-1}n^{-1/2}U_n(\beta^\star)+o_P(1). \] A martingale or independent-cluster CLT supplies the covariance \(B\), and Slutsky's theorem completes the argument.
The model-based information identity \(A=B\) holds under the correctly specified Cox intensity. Under misspecification or within-cluster dependence, a robust estimator uses empirical score-residual outer products for \(B\). This changes the variance, not the pseudo-true target \(\beta^\star\); scientific interpretation must therefore be reconsidered rather than rescued by robust standard errors alone.