Score and information of Cox partial likelihood ($U(\beta )={\sum }_j[z_{i_j}-{\bar{z}}_j(\beta )],I(\beta )={\sum }_j{\mathrm{Cov}}_{w_j}(z)$)

At an untied event time, the Cox partial likelihood weights the current risk set by $w_{jk}=e^{{\beta }^T{z}_k}/{\sum }_{\ell \in R_j}{e}^{{\beta }^T{z}_{\ell }}$. Differentiation gives the displayed score function and the negative Hessian as a sum of weighted covariances. The score is the sum of Schoenfeld functions, so Schoenfeld residuals sum to zero at a finite interior unpenalized fit. At arbitrary trial coefficients or boundary fits, a zero sum is not a general identity.