 Grouped Nelson–Aalen event-count identity

Suppose everyone enters observation at zero and has one event or censoring time. At distinct recorded time $t_j$, let $n_j$ observations terminate, of which $d_j$ are events, and use the pre-time risk set size $r_j={\sum }_{k\ge j}{n}_k$. For the grouped Nelson–Aalen estimator, $\hat{H}(t_j)={\sum }_{k\le j}{d}_k/r_k$. Exchanging sums gives ${\sum }_j{n}_j\hat{H}(t_j)={\sum }_k(d_k/r_k){\sum }_{j\ge k}{n}_j={\sum }_k{d}_k$. The unweighted sum over distinct times generally does not have this property. Processing each observation separately after breaking ties also restores the corresponding unweighted identity on the resulting artificial time grid.