 Solution

At event time $t_j$, the vector Schoenfeld residual iswhere $\hat{\beta }$ is the fitted partial-likelihood coefficient. It compares the covariate of the person who actually failed with the covariate mean predicted for the failing person, using the fitted hazard weights over the risk set. Under the model at the true coefficient, the conditional expected failing covariate is precisely that weighted mean, so the corresponding Schoenfeld function is conditionally centered at zero. Residuals are defined at observed events, not at censoring times.

The score equation from part (a) is exactlyAt a finite interior unpenalized maximum, $U(\hat{\beta })=0$, soThis is the intended fitted-residual identity. For an arbitrary trial coefficient it is a score, not generally zero; a single event with covariates zero and one in its risk set can have Schoenfeld function $1/2$ at $\beta =0$. Penalized estimates or a boundary maximum also need their own score conditions rather than this unqualified zero-sum assertion.