= Small-jump comparison of cumulative hazard estimators
{title2=$-\log(1-u)-u=\tfrac12u^2+O(u^3)$}
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 $0\le u<1$,
$$
0\le-\log(1-u)-u=\sum_{m=2}^{\infty}\frac{u^m}{m}\le\frac{u^2}{2(1-u)}.
$$
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.
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