 Small-jump comparison of cumulative hazard estimators

At a failure time with $d$ events among $r$ at risk, the Nelson–Aalen estimator adds $u=d/r$, whereas minus the logarithm of the Kaplan–Meier estimator adds $-\log (1-u)$. For $0\le u<1$,Thus small event fractions give close cumulative hazard estimates. Large risk sets alone do not suffice when a large fraction fail together. If all remaining individuals fail, the logarithmic estimate becomes infinite while the Nelson–Aalen jump is one.