Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 2 / iv

Put $u_m=d_m/r_m$. The Taylor series of the logarithm gives the small-jump comparison of cumulative hazard estimators:Consequently, if ${\max }_m{u}_m\le \epsilon <1$, the two integrated hazard estimates obeyThey 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.