d_{j} / r_{j}

t_{j}

H_{0} (t) = \sum_{t_{j} \leq t} \frac{d _{j}}{\sum _{i \in R (t_{j})} ϕ _{i}} .

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)} ϕ_{i}

; with every multiplier equal to one it reduces to the Nelson–Aalen estimator.

The estimator is right-continuous, so

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}

. Hence

H_{0} (t_{d}) - H_{0} (t_{a}) = \frac{1}{ϕ _{c} + ϕ _{d}} + \frac{1}{ϕ _{d}} .

There is no increment at the censoring time

t_{b}

. If the lower endpoint had instead been

t_{a} -

, an additional

1/ (ϕ_{a} + ϕ_{b} + ϕ_{c} + ϕ_{d})

would be present. The final singleton event can increase the baseline hazard estimate even though its partial likelihood contribution is one.

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