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For subject $i$, write ${\delta }_i=1_{\{T_i\le C_i\}}$, and define the counting process and pre-event at-risk process byUnder independent censoring and a common hazard, each observed event-free subject has conditional event probability $h(t)dt$ in a short interval. Therefore $\mathbb{E}(dN(t)\mid \text{past})=R(t)dH(t)$ to first order. Equivalently the compensator of a counting process is ${\int }_0^{t}R(s)dH(s)$. Dividing the observed increments by current exposure gives the estimating equation $d\hat{H}=dN/R$, henceHere $t_j$ are distinct event times, $d_j$ the number of events at that time, and $r_j=R(t_j)$ the number at risk immediately before it. One can also obtain the same increment by maximizing the local multiplicative-intensity likelihood $a^{d_j}{e}^{-r_{j}a}$: differentiation gives $a=d_j/r_j$. This derives the Nelson–Aalen estimator, rather than merely naming it.

A censored observation contributes to the denominator while under observation, and then leaves the risk set; it contributes no event jump. For tied events, use the single increment $d_j/r_j$, not $d_j$ successive one-event increments with shrinking denominators. If censoring and events share a recorded time, a convention is needed; counting the event before removing same-time censored subjects uses the pre-time risk set above. Rounded or interval-censored data may require a model suited to their observation mechanism.

For the distinct ordered observations in the question, let ${\delta }_j$ indicate whether $X_j$ is an event. The risk set at $X_j$ contains precisely the $n-j+1$ subjects whose recorded time is at least $X_j$. ThusTherefore the sum of the fitted integrated hazards at all observation times equals the observed number of events, the event-count identity for Nelson–Aalen cumulative hazards. The grouped Nelson–Aalen event-count identity extends the same argument to tied observations with their multiplicities and the stated risk-set convention.

Use the standard right censoring setup with everyone entering observation at time zero. At each observed time $x_j$, the pre-exit risk set has size $r_j=\#\{i:x_i\ge x_j\}$. The Nelson–Aalen estimator of the cumulative hazard function isCensoring removes an individual from later risk sets but contributes no event increment.

Exchange the two finite sums:Each event increment appears in exactly as many subjects' fitted cumulative hazards as there were subjects at risk for that increment. This proves the event-count identity for Nelson–Aalen cumulative hazards. Common entry is relevant: with delayed entry, a subject does not accumulate the increments before entering, so an entry-time subtraction would be needed.

One method for ties is to group observations at each distinct recorded time. If $d_k$ events occur at ${\tau }_k$, use the pre-time risk set size $R_k$ including observations censored at that same recorded time, and take one jump $d_k/R_k$. This is the usual events-before-censoring convention within a recorded tie. Since $R_k=\#\{i:x_i\ge {\tau }_k\}$,Thus the grouped Nelson–Aalen event-count identity preserves the required identity at the recorded observation times. It is a sum over individuals, not an unweighted sum over distinct times.

A second method, suitable when ties reflect rounded times, is to break ties by small perturbations or an artificial ordering, then apply the untied estimator. The event-count identity holds on that new, untied time grid. It does not generally persist if all estimates are mapped back to the original common time. For example, two tied failures among two individuals give a grouped jump $2/2=1$, so the original-time sum is $2$. Sequentially breaking the failures gives jumps $1/2$ and $1$, whose total is $3/2$; assigning the completed jump to both original tied observations gives sum $3$. On the perturbed grid the two values are $1/2$ and $3/2$, whose sum is again $2$. This distinction states exactly what the second method does and does not preserve.

Another common grouped survival approach is the Kaplan–Meier estimator, from which one may estimate cumulative hazard by $-\log \hat{S}(t)$. Its jump is $-\log (1-d_k/R_k)$, not $d_k/R_k$, and it likewise does not have the Nelson–Aalen event-count identity. In particular it becomes infinite at a time when all remaining individuals fail.

One method breaks a tied group into a sequence of distinct artificial times, by a specified or random ordering. Process tied events before tied censorings if their common recorded time means the censored individuals were still under observation when the events occurred. At each artificial event, add the reciprocal of the current risk set size and then remove that individual. This applies the untied Nelson–Aalen estimator to the resulting grid. For example, two tied failures among two individuals give successive increments $1/2$ and $1$.

On that expanded grid there is one observation time per individual. The cancellation argument in (b) is unchanged, so breaking ties preserves the identity when the sum includes every artificially separated observation. The chosen ordering can affect the estimated curve within a tied group and must not be presented as an observed event order.

A second method retains the recorded tied times. For a group with $d_j$ events and pre-time risk set size $r_j$, the grouped Nelson–Aalen estimator adds $d_j/r_j$ once. All events in the group use the same denominator; censorings in the group are removed afterwards. If two individuals fail at the same time and there are no other observations, the curve has one jump of $2/2=1$. Its sum over the one distinct time is $1$, whereas the event count is $2$. Hence the unweighted distinct-time identity does not generally survive grouping ties. On the artificially separated grid in the first method, the two post-observation values would instead be $1/2$ and $3/2$, whose sum is $2$.

There is nevertheless an exact weighted version for the grouped method. Let $n_j$ count all observations, event or censoring, at distinct time $t_j$. With everyone entering at zero, $r_k={\sum }_{j\ge k}{n}_j$. ThereforeThis grouped Nelson–Aalen event-count identity sums over individuals, with repeated values for tied times, rather than once per distinct time. Keeping that distinction resolves the apparent conflict between a valid grouped estimator and failure of the literal unweighted formula.