 Solution

The difference $s(t^{\ast },\beta )$ is the observed event subject's covariate minus its conditional expected value under the working coefficient. It is the Schoenfeld function; at a fitted coefficient it becomes a Schoenfeld residual. If the conditional event-label distribution is generated by coefficient $\beta$, part (b) givesHence its unconditional expectation is zero whenever it is integrable. Centering is at the true coefficient: data generated under ${\beta }_0$ do not generally make $s(t^{\ast },\beta )$ mean zero at an arbitrary trial value $\beta \ne {\beta }_0$.

The same quantity is the score function of one conditional event factor, sinceThis connects its conditional centering to the usual zero-mean likelihood-score identity.