Cox partial-likelihood score expansion and estimator linearization

Assume AC. Let $p<\mathrm{\infty }$, $\tau >0$, and ${\beta }_0\in {\mathbb{R}}^p$. Let subjects $O_i=(T_i,C_i,Z_i(\cdot ))$ be IID and independent. Assume the Cox model of Cox proportional-hazards model and partial likelihood with bounded predictable covariates $\mathrm{\parallel }{Z}_i(t){\mathrm{\parallel }}_2\le K$, baseline ${\lambda }_0\in L^1[0,\tau ]$, and no tied failures. Assume the positive population risk denominators of Integrated and conditional uniform convergence of Cox risk-set sums on a compact neighborhood $B$ of ${\beta }_0$. In one common filtration, assume the observed event counts $N_i(t)=1\{U_i\le t,{\delta }_i=1\}$ have compensators $A_i(t)={\int }_0^t{Y}_i(u)e^{{\beta }_0^T{Z}_i(u)}{\lambda }_0(u)du$ and that $M_i=N_i-A_i$ satisfies the square-integrable counting-process contract of Simple counting-process intensity and compensator contract. This martingale contract is an explicit model assumption. No bracketing or time-class regularity is required: the risk-set lemma supplies the integrated parameter-uniform laws for these bounded predictable paths.

Let $I$ be the matrix in Uniform Cox partial-likelihood Hessian convergence and assume $I$ is positive definite. Let ${\widehat{\beta }}_n$ be a measurable ${\mathbb{R}}^p$-valued estimator such that, with probability tending to one, it solves the partial-likelihood score equation $U_n({\widehat{\beta }}_n)=0$, and suppose ${\widehat{\beta }}_n{\to }_P{\beta }_0$. Then $\sqrt{n}({\widehat{\beta }}_n-{\beta }_0)=I^{-1}\frac{1}{\sqrt{n}}{\sum }_{i=1}^n{\int }_0^{\tau }\{Z_i(t)-\bar{z}(t)\}d{M}_i(t)+o_P(1),\bar{z}(t)=\frac{s^{(1)}(t,{\beta }_0)}{s^{(0)}(t,{\beta }_0)}\mathrm{.}$ The proof establishes the partial-likelihood score representation locally; it does not treat the score summands with the random risk-set mean as IID.