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For individual $i$, let $N_i(t)$ count its observed event by time $t$ and let $Y_i(t)$ indicate membership of the risk set immediately before $t$. Set $N={\sum }_i{N}_i$ and $Y={\sum }_i{Y}_i$. Under a common hazard function $h(t)$ and independent censoring, a currently at-risk subject has event probability $h(t)dt+o(dt)$ in the next short interval. Summing over the current risk set gives the conditional intensityEquivalently, the event count decomposes aswhere $M$ is the mean-zero counting-process martingale obtained by subtracting its conditional compensator. Estimating the unknown hazard increment by the observed event increment divided by the risk set size gives the Nelson–Aalen estimatorHere $a_j$ are the distinct observed event times and $r_j=Y(a_j)$ includes the subject who is about to fail. A censored observation affects subsequent risk sets, but causes no event jump. This is the requested no-ties estimator; tied event counts would produce $d_j/r_j$.

The derivation also identifies the estimation error:Thus it estimates $H(t)$ on the time range where individuals remain at risk. It is not exactly unbiased for the entire cumulative hazard after the last risk set disappears. Its usual Nelson–Aalen variance estimator is ${\sum }_{a_j\le t}{r}_j^{-2}$, since the compensator variance increment of the no-ties event process is $YdH$.

Put $q_t=\mathrm{Pr}(d{N}_{+}(t)=1\mid {\mathcal{H}}_{t-})$. A binary event increment has conditional variance $q_t(1-q_t)$. Since $q_t=Y_{+}(t)h(t)dt+o(dt)$, its squared mean is of second order, givingTreating the predictable risk set size as known given the history, the Nelson–Aalen estimator increment therefore has first-order varianceSubstitute $d\hat{H}=d{N}_{+}/Y_{+}$ for $dH$ to obtain the estimated increment variance $d{N}_{+}/Y_{+}^2$. Summing these estimated predictable variances gives the usual Nelson–Aalen variance estimator:The martingale $d{N}_{+}-Y_{+}dH$ has orthogonal increments, which justifies accumulating the predictable variances of the estimation error. This is an estimated sampling variance on the observed at-risk range, not a claim of exact finite-sample unbiasedness after the risk set becomes empty. For multiple events at an event time the usual extension replaces the numerator 1 by the event count.