Nelson–Aalen estimator ($\hat{H}(t)$)

The Nelson–Aalen estimator of the cumulative hazard is $\hat{H}(t)={\sum }_{t_j\le t}{d}_j/r_j$, where $d_j$ events occur among $r_j$ individuals at risk at event time $t_j$.

At a failure time with $d$ events among $r$ at risk, the Nelson–Aalen estimator adds $u=d/r$, whereas minus the logarithm of the Kaplan–Meier estimator adds $-\log (1-u)$. For $0\le u<1$,Thus small event fractions give close cumulative hazard estimates. Large risk sets alone do not suffice when a large fraction fail together. If all remaining individuals fail, the logarithmic estimate becomes infinite while the Nelson–Aalen jump is one.

Suppose individual event hazards have the additive form $h_i=h_{0i}+h_1$, where $h_{0i}$ is known and $h_1$ is common. The aggregate counting-process intensity in survival analysis is $\lambda ={\sum }_i{Y}_i{h}_{0i}+Y{h}_1$, with $Y={\sum }_i{Y}_i$ the current risk set size. Writing $dN=\lambda dt+dM$ and dividing by $Y$ givesThe last term is a counting-process martingale integral. This establishes the cumulative common-hazard target on the observed at-risk time range. The known offset must be averaged over the current risk set, not the initial sample. Between event jumps, its subtraction can make the unconstrained estimate decrease. If all known hazards equal $h_0$ and $Y>0$ up to $t$, the estimate reduces to the Nelson–Aalen estimator minus $H_0(t)$.

With all subjects entering at time zero and no tied observation times, the Nelson–Aalen estimator satisfies ${\sum }_i\hat{H}(x_i)=d$. Exchanging sums makes each event increment $1/Y(t_j)$ appear in exactly $Y(t_j)$ subject hazards. The identity relies on the risk-set count equalling the number of observation times at least $t_j$, and generally needs modification for delayed entry.

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.

For untied events at $a_j$ with predictable risk set size $Y_j$, the estimated predictable variation of the Nelson–Aalen estimator is ${\sum }_{a_j\le t}{Y}_j^{-2}$. It comes from the martingale event increment variance $YdH$ and the estimator increment $dN/Y$. It applies on the range with individuals at risk; it is not an exact finite-sample unbiasedness statement.

Even with a common event hazard, the exact expectation of the Nelson–Aalen estimator is ${\int }_0^t\mathbb{P}(Y(u)>0)h(u)du$, because no increments can be observed after the risk set is empty. Replacing ratios by ratios of expectations gives an approximate target and does not prove finite-sample unbiasedness. For one uncensored exponential event time, the estimator has mean $1-e^{-\lambda t}$ rather than $\lambda t$.