Uniform Cox partial-likelihood Hessian convergence

At each recorded failure the risk set contains its failing subject, so its denominator is positive for every $\beta$, even if risk sets are empty at other times. Differentiate the finite sum defining $\log L_p$ to obtain $U_n(\beta )={\sum }_i\int (Z_i-{\bar{Z}}_n(t,\beta ))d{N}_i$ and $H_n(\beta )=-\int V_n(t,\beta )d{N}_{\cdot }$. Define $V_n=0$ on empty risk sets. It is predictable, its entries are bounded by $2K^2$, and [F5] permits the pathwise decomposition against $d{N}_{\cdot }=d\mathcal{M}+d{A}_{\cdot }$.

Since subjects have no simultaneous failures, $N_{\cdot }$ is simple; $\mathcal{M}={\sum }_i{M}_i$ is square-integrable and has continuous compensator $A_{\cdot }={\sum }_i{A}_i$. Its total compensator satisfies $A_{\cdot }(\tau )\le n{e}^{K\mathrm{\parallel }{\beta }_0{\mathrm{\parallel }}_2}\mathrm{\parallel }{\lambda }_0{\mathrm{\parallel }}_1$, and $E{N}_{\cdot }(\tau )=E{A}_{\cdot }(\tau )$ by [F5]. Thus the aggregate contract holds and [L2]–[L3] apply. For each fixed $\beta$, including the random predictable $V_n(t,\beta )$, their isometry bounds $E\mathrm{\parallel }{n}^{-1}\int V_n(t,\beta )d\mathcal{M}{\mathrm{\parallel }}_F^2\le C\mathrm{/}n$.

The parameter maps $V_n(t,\beta )$ have a deterministic Lipschitz bound independent of $n,t$: on a nonempty risk set, differentiating a weighted expectation of a bounded coordinate $g(Z)$ gives $E_{n,\beta }[gZ]-E_{n,\beta }[g]E_{n,\beta }[Z]$, bounded by $2K\sup \mathrm{\mid }g\mathrm{\mid }$; apply this to $g=Z_j,Z_j{Z}_k$ and the covariance product. An empty risk set is empty for every parameter, so its zero convention also has this bound. The population maps satisfy the same bounds by differentiation under their bounded expectations and positive denominators. A finite $\epsilon$-net of $B$ and step 2.1 control $n^{-1}\int V_{n}d\mathcal{M}$ on net points. Between points, [F5] bounds the difference by $C\epsilon (N_{\cdot }(\tau )+A_{\cdot }(\tau ))\mathrm{/}n=O_P(\epsilon )$; letting $\epsilon \downarrow 0$ proves its uniform parameter convergence to zero. The population Lipschitz bound and integrability of $s^{(0)}(t,{\beta }_0){\lambda }_0(t)$ also prove continuity of $J$.

The compensator term equals $\int V_n(t,\beta )S_n^{(0)}(t,{\beta }_0){\lambda }_0(t)dt$. Its difference from $J(\beta )$ is bounded uniformly in parameter by $C{\int }_0^{\tau }\{{\sup }_{\beta \in B}\mathrm{\parallel }{V}_n(t,\beta )-V(t,\beta ){\mathrm{\parallel }}_{\max }+\mathrm{\mid }{S}_n^{(0)}(t,{\beta }_0)-s^{(0)}(t,{\beta }_0)\mathrm{\mid }\}{\lambda }_0(t)dt\mathrm{.}$ [F6] and [F2]'s expected integrated squared-error convergence make this bound tend to zero in L¹ and hence probability (if $\mathrm{\parallel }{\lambda }_0{\mathrm{\parallel }}_1=0$, the bound is zero directly). Combine this with step 3.1 and the Hessian identity in step 1.1 to prove the stated uniform parameter limit. Finally $J({\beta }_0)=I$ by direct multiplication of $V(t,{\beta }_0)$ by $s^{(0)}(t,{\beta }_0)$.