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The compensator of a counting process $N(t)$ is a predictable increasing process $\Lambda (t)$ such that $N(t)-\Lambda (t)$ is a local martingale. For an observed event process with at-risk process $Y(t)$ and hazard function $h(t)$, it is $\Lambda (t)={\int }_0^{t}Y(u)h(u)du$ under independent censoring. It is cumulative intensity, rather than an event probability.

An observed event counting process has conditional increment $\mathbb{E}(d{N}_i(t)\mid {\mathcal{H}}_{t-})=Y_i(t)h_i(t)dt$, where $Y_i$ is the at-risk process. The filtration records the past observed events, censoring, and covariates; the integral of $Y_i{h}_i$ is the compensator of a counting process.

Let $N(t)$ count observed failures and let $Y(t)$ be the at-risk process, counting subjects under observation and still alive immediately before $t$. Under independent censoring, a short interval of length $dt$ has conditional expected failure count $Y(t)h(t)dt=Y(t)dH(t)$. The estimating equation obtained by replacing the count expectation by its observation is $d\hat{H}(t)=dN(t)/Y(t)$. Each distinct failure gives $\Delta N(a_m)=1$ and $Y(a_m)=r_m$, so the Nelson–Aalen estimator isThe same increment maximizes the local working likelihood for a Poisson distribution with mean $r_m\Delta H_m$: differentiating $\Delta N_m\log (\Delta H_m)-r_m\Delta H_m$ gives $\Delta H_m=\Delta N_m/r_m$. The counting-process derivation explains why the risk set, rather than the initial sample size, supplies the denominator.

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.

Keep the counting process and at-risk process notation from the introductory derivation. Because the individual hazards add, the aggregate conditional intensity isThe known individual contributions must be averaged over the current risk set, rather than over the original cohort. DefineThen $dN(t)/Y(t)$ estimates $\{{\overline{h}}_0(t)+h_1(t)\}dt$. Subtract the known average contribution to obtain the offset-adjusted cumulative hazard estimatorTake the quotient to be zero when $Y=0$. If $M(t)=N(t)-{\int }_0^t\lambda (u)du$, direct substitution givesThis establishes the target and the mean-zero estimation noise, under the usual integrability conditions. It estimates $H_1(t)$ while there is a nonempty risk set; no data identify its continuation after follow-up has ended. With common known $h_{0i}=h_0$ and individuals at risk throughout $[0,t]$, it reduces to ${\hat{H}}_1(t)=\hat{H}(t)-H_0(t)$.

The estimate may decrease between events because the known offset is subtracted continuously. That is not an algebraic error: it is an unconstrained estimating-equation estimator, not automatically a nonnegative monotone cumulative hazard estimate. If such shape constraints are imposed, the fitting procedure must explicitly account for them. The working hazard model itself must satisfy $h_{0i}+h_1\ge 0$.

The history ${\mathcal{H}}_{t-}$ contains the information observed strictly before $t$: baseline characteristics, previous events and censoring, and therefore which individuals are currently eligible and under observation. Formally it is the pre-$t$ information in the relevant filtration. The at-risk process $Y_i(t)$ is 1 if individual $i$ is observed and event-free immediately before $t$, and 0 otherwise. Let $Y_{+}(t)={\sum }_i{Y}_i(t)$ and write $dH(t)=h(t)dt$ for the cumulative hazard increment.

The counting-process intensity in survival analysis givesThe common conditional hazard function is assumed to remain applicable after conditioning on the observed history, as under suitable independent censoring. Summing gives the conditional mean of the total event increment:On times with $Y_{+}(t)>0$, invert this relation to estimate the hazard increment:The last sum is for untied events. This is the Nelson–Aalen estimator; for ties replace 1 by the number of events at that time. No increment can be estimated once the risk set is empty.

With independent censoring, the at-risk process is $Y_i(t)=1_{T_i\ge t,C_i\ge t}$. Independence and the common censoring survivor function $G$ give

In the counting-process intensity in survival analysis, the counting process $N_i(t)$ jumps once if an observed event occurs; the at-risk process $Y_i(t)$ is one immediately before $t$ only while individual $i$ is still observed and event-free. Conditional on that information, the chance of an event in a short interval is $h_i(t)dt+o(dt)$ for an at-risk individual and zero otherwise. This is the meaning ofThe history ${\mathcal{H}}_{t-}$ is the filtration generated by the observed event and censoring histories, entry information, and available covariates strictly before $t$. In particular, it determines the current risk set but does not reveal future event times. The right-hand side is the infinitesimal compensator of a counting process; the equality is interpreted as an intensity statement to first order in $dt$.

Define the event counting process and at-risk process byUnder a common hazard function and independent censoring, the conditional event rate is $Y(u)h(u)$. Equivalently, the counting-process compensator satisfies $\mathbb{E}[dN(u)\mid {\mathcal{F}}_{u-}]=Y(u)dH(u)$. Estimate a cumulative hazard increment by the observed event increment divided by the current risk set:With no ties, $dN$ has mass $v_i$ at $x_i$, and $Y(x_i)={\sum }_{j:x_j\ge x_i}1$. Thus the Nelson–Aalen estimator isOnly event times contribute, so no empty risk set is divided into an event count. Right-censored observations contribute time to the risk sets, but no event increment.

$N(t)$ is the event counting process, while $Y(t)$ is the at-risk process. Their key relationship is that, conditionally on the observed past, the expected event increment is $Y(t)dH(t)$. It separates exposure to risk from event occurrence and naturally incorporates right censoring.

Before the observed endpoint, $Y+N=1$. If censoring occurs before the event, both processes are zero after the censoring time, so their sum is zero. With the inclusive endpoint conventions in the question, at an observed event time $t=T\le C$ both are one, so the sum is two at that single instant.

Let $Y(t)$ be the at-risk process and $N(t)$ the counting process for observed events. Over a short interval, the multiplicative-intensity model giveswhere $h$ is the hazard function and $H$ the cumulative hazard function. Solving this relation for the infinitesimal hazard increment suggests $d\hat{H}(t)=dN(t)/Y(t)$. Summing over distinct event times gives the Nelson–Aalen estimatorwhere $d_j$ events occur among $r_j$ individuals at risk. Here there are no ties, so $d_j=1$.