s (t^{*}, β)

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

β

, part (b) gives

E_{β} [s (t^{*}, β) ∣ F_{t^{*} -}, one event] = E_{β} [z_{π (t^{*})} ∣ \dots] - \overline{z} (t^{*}, β) = 0.

Hence its unconditional expectation is zero whenever it is integrable. Centering is at the true coefficient: data generated under

β_{0}

do not generally make

s (t^{*}, β)

mean zero at an arbitrary trial value

β \neq = β_{0}

The same quantity is the score function of one conditional event factor, since

\frac{\partial}{\partial β} lo g \frac{e ^{β z_{π (t^{*})}}}{S _{0} ( t ^{*} , β )} = z_{π (t^{*})} - \frac{S _{1} ( t ^{*} , β )}{S _{0} ( t ^{*} , β )} .

This connects its conditional centering to the usual zero-mean likelihood-score identity.

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