= Solution
Put $u_m=d_m/r_m$. The <Taylor series> of the logarithm gives the <small-jump comparison of cumulative hazard estimators>:
$$
-\log(1-u_m)=u_m+\frac{u_m^2}{2}+\frac{u_m^3}{3}+\cdots.
$$
Consequently, if $\max_m u_m\le\varepsilon<1$, the two <integrated hazard> estimates obey
$$
0\le\widehat H_{\rm KM}-\widehat H_{\rm NA}\le\frac1{2(1-\varepsilon)}\sum_m u_m^2\le\frac{\varepsilon}{2(1-\varepsilon)}\widehat H_{\rm NA}.
$$
\b[They are close when every event fraction is small.] In particular, with distinct single failures and large <risk sets>, $u_m=1/r_m$ is small. With tied data, large <risk sets> alone are insufficient if a substantial fraction of the <risk set> fails together.
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