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

Collect distinct observed failure times as $a_m$, with $d_m$ failures and $r_m$ individuals in the risk set just before $a_m$. The grouped Nelson–Aalen estimator isAll failures recorded at one time use the same pre-event risk set. When failures and censoring share a recorded time, the usual convention includes those censored at that time in the risk set and processes failures before removing censored individuals. A known different ordering should instead be honored. If ties arise solely through rounding of a continuous-time process, an alternative is to resolve the unobserved ordering: $d_m$ successive failures without intervening censoring would contribute ${\sum }_{s=0}^{d_m-1}(r_m-s{)}^{-1}$. That is not the grouped jump $d_m/r_m$; the difference represents a choice about what the recorded ties mean.