Nelson–Aalen, Kaplan–Meier, and Greenwood estimators

For a positive integer sample size $n$, let $(T_i,C_i)$, $1\le i\le n$, be event and censoring times with right-censored observations $(U_i,{\delta }_i)$ as in Event-time observation, censoring, and truncation schemes, so $U_i=\min (T_i,C_i)$ and ${\delta }_i=\mathbf{1}\{T_i\le C_i\}$. The subject counting process and at-risk process are $N_i(t)=\mathbf{1}\{U_i\le t,{\delta }_i=1\}$ and $Y_i(t)=\mathbf{1}\{U_i\ge t\}$. The endpoint convention makes $Y_i$ left-continuous at $U_i$: a subject is still at risk at the observed time, including a tied event and censoring time. Set $N_n={\sum }_{i=1}^n{N}_i$ and $Y_n={\sum }_{i=1}^n{Y}_i$. Let $d{N}_n$ be the finite counting measure of observed events on $[0,\mathrm{\infty })$, with atom $\mathrm{\Delta }{N}_n(u):={\sum }_{i=1}^n\mathbf{1}\{U_i=u,{\delta }_i=1\}$ at each $u\ge 0$; in particular, $\mathrm{\Delta }{N}_n(0)=N_n(0)$. Tied failures at $u$ contribute their full count and censorings at $u$ do not contribute to $d{N}_n(u)$.

For $t\ge 0$, the Nelson–Aalen estimator is ${\widehat{\mathrm{\Lambda }}}_n(t)={\int }_{[0,t]}\frac{\mathbf{1}\{Y_n(u)>0\}}{Y_n(u)}d{N}_n(u)$, where the integrand is defined to be zero when $Y_n(u)=0$. The Kaplan–Meier estimator is ${\widehat{S}}_n(t)={\prod }_{\begin{array}{c}0\le u\le t:\mathrm{\Delta }{N}_n(u)>0\end{array}}(1-\frac{\mathrm{\Delta }{N}_n(u)}{Y_n(u)})$. Each product is finite and a factor is zero if every subject at risk fails at that time. The target survival function is $S(t)=P(T>t)$. If $T$ has density $f$, $S$ is positive on the horizon, and $\lambda =f\mathrm{/}S$, the continuous cumulative hazard is $\mathrm{\Lambda }(t)={\int }_0^t\lambda (u)du=-\log S(t)$.

These inclusive endpoint formulas reduce to the usual $(0,t]$ formulas whenever $\mathrm{\Delta }{N}_n(0)=0$. If $T$ has finite support $0\le a_1<\cdots <a_m$, write $q_j=P(T=a_j\mid T\ge a_j)$; the corresponding discrete cumulative hazard is ${\mathrm{\Lambda }}_d(t)={\sum }_{a_j\le t}{q}_j$. Under IID sampling, independent censoring, and positive risk probability at the support points, the Nelson–Aalen increments estimate these $q_j$; in general this discrete cumulative hazard is not $-\log S(t)$.

A baseline event is included at $u=0$; if it occurs then $N_i(0)=1$, so a model contract requiring $N_i(0)=0$ must exclude this case.

An empty event product is $1$, and an empty Greenwood event sum is $0$.

On a horizon where every event contribution has $Y_n(u)-\mathrm{\Delta }{N}_n(u)>0$, the Greenwood variance estimate is ${\widehat{V}}_G(t)={\widehat{S}}_n(t{)}^2{\sum }_{0\le u\le t:\mathrm{\Delta }{N}_n(u)>0}\frac{\mathrm{\Delta }{N}_n(u)}{Y_n(u)\{Y_n(u)-\mathrm{\Delta }{N}_n(u)\}}$. Greenwood's formula is used as a first-order asymptotic variance estimate when the sampling and censoring assumptions justify that approximation; the formula alone asserts no asymptotic result and is not an exact finite-sample variance identity. It is undefined when a contributing denominator is zero.

For each fixed $t$, ${\widehat{\mathrm{\Lambda }}}_n(t)$ and ${\widehat{S}}_n(t)$ are statistics of the observed sample in the sense of Statistics and estimators. Greenwood's formula is a measurable function on the measurable subset $D_n(t)$ of the observed sample space $([0,\mathrm{\infty })\times \{0,1\}{)}^n$ where every contributing denominator is positive. Its zero extension ${\widehat{V}}_G^{\circ }(t)$, equal to ${\widehat{V}}_G(t)$ on $D_n(t)$ and $0$ elsewhere, is a statistic on the whole observed sample space. This extension assigns no Greenwood variance interpretation outside $D_n(t)$.

To verify measurability, scan indices $i=1,\dots ,n$ and include only the first event observation at each distinct time in $[0,t]$; each inclusion condition uses finitely many Borel equalities and inequalities among the observed $U_i$ and ${\delta }_i$. Risk and event counts are finite Borel sums. The finitely many positive-denominator tests make $D_n(t)$ measurable; evaluating divisions only on their positive-denominator branches makes the sums, products, and zero extension Borel measurable.