Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 5 / b / iv / Solution

In the Cox proportional-hazards model, replace the number at risk in the Nelson–Aalen estimator by the sum of relative hazards:The total event counting process has conditional rate $W_{\beta }(u)h_0(u)$, so the estimated baseline hazard increment is $dN(u)/W_{\hat{\beta }}(u)$. Integrating gives the Breslow estimatorWhen $\hat{\beta }=0$, all relative hazards are 1 and this reduces to the Nelson–Aalen estimator. The denominator is positive at each observed event time.