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Use the right-continuous Breslow estimator, so the increase from the value at $x_{m-1}$ to the value at $x_m$ means the interval $(x_{m-1},x_m]$. There is no event between these consecutive exit times, and at $x_m$ the risk set contains only subject $m$. Henceand the fitted individual cumulative hazard increases byIt is the subject's integrated hazard, rather than the reference baseline increment, that equals the event indicator.

For the final unheaded request, a fitted martingale residual at a subject's exit isThe revised observation leaves the last subject at risk until a later event time, still after every other subject has exited. Every earlier risk set is exactly the same as before, and the new final event again contributes the constant factor one. Thus part a gives unchanged coefficients, and every earlier Breslow estimator increment is unchanged. No additional events occur during the extended interval because all other subjects have already left observation.

For $i<m$, the event indicator and fitted cumulative hazard at $x_i$ remain unchanged, hence so does ${\hat{M}}_i$. For the terminal subject, let $C$ denote its old fitted integrated hazard at its censored exit. That old residual is $-C$. Part b shows that the new final event adds one to its fitted integrated hazard, giving new residualTherefore all fitted martingale residuals remain unchanged. Their invariance does not assert that the fitted survivor function at later times or the terminal event record is unchanged: the new late baseline hazard jump is real. This is the terminal-observation invariance of Cox martingale residuals, which relies on a unique final subject and unchanged covariate history.