 Solution

The ordinary Nelson–Aalen estimator adds $d_j/r_j$ at event time $t_j$. Under a fitted proportional hazards model, replace the number at risk by the sum of fitted hazard multipliers:This is the Breslow estimator of the integrated baseline hazard, using fitted regression coefficients obtained by maximizing the Cox partial likelihood. The denominator arises because the total event intensity is $h_0(t){\sum }_{i\in R(t)}{\varphi }_i$; with every multiplier equal to one it reduces to the Nelson–Aalen estimator.

The estimator is right-continuous, so ${\hat{H}}_0(t_a)$ already includes the event at $t_a$. Subtracting it leaves only the events at $t_c$ and $t_d$, whose risk sets are $\{c,d\}$ and $\{d\}$. HenceThere is no increment at the censoring time $t_b$. If the lower endpoint had instead been $t_a-$, an additional $1/({\hat{\varphi }}_a+{\hat{\varphi }}_b+{\hat{\varphi }}_c+{\hat{\varphi }}_d)$ would be present. The final singleton event can increase the baseline hazard estimate even though its partial likelihood contribution is one.