The Kaplan-Meier estimator reconstructs survival by multiplying empirical conditional survival probabilities. The Nelson-Aalen estimator instead adds empirical hazard increments. Their finite-sample forms are elementary; their rigorous large-sample theory is most transparent through the counting-process martingales introduced on the next page.
Risk sets and observed failures
Definition. For independently right-censored observations \((X_i,\Delta_i)\), define \[ N_i(t)=1\{X_i\le t,\Delta_i=1\},\qquad Y_i(t)=1\{X_i\ge t\}, \] and \(N(t)=\sum_iN_i(t)\), \(Y(t)=\sum_iY_i(t)\). If \(t_1<\cdots<t_m\) are the distinct observed failure times, put \[ d_j=\Delta N(t_j),\qquad r_j=Y(t_j). \] With delayed entry \(A_i\), replace \(Y_i(t)\) by \(1\{A_i\le t\le X_i\}\).
The convention \(Y_i(t)=1\{X_i\ge t\}\) makes a subject failing at \(t\) a member of the risk set immediately before \(t\).
The product-limit likelihood
Theorem (Kaplan-Meier NPMLE). Assign a conditional failure probability \(q_j=P(T=t_j\mid T\ge t_j)\) at each observed failure time. The nonparametric likelihood is \[ L(q_1,\ldots,q_m) =\prod_{j=1}^m q_j^{d_j}(1-q_j)^{r_j-d_j}, \] up to factors not involving \(q\). Its maximizer is \(\widehat q_j=d_j/r_j\), and the resulting product-limit estimator is \[ \widehat S(t) =\prod_{t_j\le t}\left(1-\frac{d_j}{r_j}\right). \] If the largest observation is censored, the mass remaining beyond it is not identified; \(\widehat S\) is nevertheless uniquely determined before that endpoint.
Proof. A subject failing at \(t_j\) contributes \(q_j\) after surviving all earlier event times. A subject known to survive through \(t_j\) contributes \(1-q_j\). Regrouping all individual contributions at each \(t_j\) gives the displayed binomial likelihood. Its log is \[ \ell_j(q_j)=d_j\log q_j+(r_j-d_j)\log(1-q_j), \] which is strictly concave and has maximizer \(q_j=d_j/r_j\) when \(0<d_j<r_j\), with boundary cases obtained continuously. Multiplication of the estimated conditional survival probabilities yields \(\widehat S\).
Example. Suppose the ordered observations are \[ (2,1),(3,0),(4,1),(5,1),(6,0). \] At failures \(2,4,5\), the risk-set sizes are \(5,3,2\). Therefore \[ \widehat S(t)= \begin{cases} 1,&t<2,\\ 4/5,&2\le t<4,\\ (4/5)(2/3)=8/15,&4\le t<5,\\ (4/5)(2/3)(1/2)=4/15,&t\ge5. \end{cases} \] The censored observation at time \(3\) reduces later risk sets but causes no survival jump.
Nelson-Aalen cumulative hazard
Definition. The Nelson-Aalen estimator is \[ \widehat\Lambda(t) =\int_{(0,t]}\frac{1\{Y(u)>0\}}{Y(u)}\,\mathrm{d}N(u) =\sum_{t_j\le t}\frac{d_j}{r_j}. \] Its product-integral transform \[ \prod_{(0,t]}\{1-\mathrm{d}\widehat\Lambda(u)\} \] equals \(\widehat S(t)\) exactly.
Proposition. If \(\max_{t_j\le t}d_j/r_j\le c<1\), then \[ 0\le-\log\widehat S(t)-\widehat\Lambda(t) \le \frac{1}{2(1-c)} \sum_{t_j\le t}\left(\frac{d_j}{r_j}\right)^2. \]
Proof. For \(0\le x\le c<1\), \[ 0\le-\log(1-x)-x =\sum_{k\ge2}\frac{x^k}{k} \le \frac{x^2}{2(1-x)} \le\frac{x^2}{2(1-c)}. \] Apply this inequality to every hazard jump and sum.
Martingale representation and consistency
Theorem. Assume independent censoring and a continuous event hazard \(\lambda\). Then \[ M_i(t)=N_i(t)-\int_0^tY_i(u)\lambda(u)\,\mathrm{d}u \] is a martingale, and on any interval on which \(Y(u)>0\), \[ \widehat\Lambda(t)-\Lambda(t) =\int_0^t\frac{1}{Y(u)}\,\mathrm{d}M(u), \qquad M=\sum_{i=1}^nM_i. \] If \(Y(t)/n\to y(t)\) uniformly in probability on \([0,\tau]\), with \(\inf_{t\le\tau}y(t)>0\), then \[ \sup_{t\le\tau}|\widehat\Lambda(t)-\Lambda(t)| \xrightarrow{P}0. \]
Proof. Summing \(\mathrm{d}N_i=Y_i\lambda\,\mathrm{d}t+\mathrm{d}M_i\) gives \[ \mathrm{d}N=Y\lambda\,\mathrm{d}t+\mathrm{d}M. \] Divide by \(Y\) and integrate to obtain the exact representation. Its predictable quadratic variation is \[ \left\langle\widehat\Lambda-\Lambda\right\rangle(t) =\int_0^t\frac{\lambda(u)}{Y(u)}\,\mathrm{d}u=O_P(n^{-1}). \] Lenglart's or Doob's inequality makes the martingale supremum \(o_P(1)\). The event that \(Y\) vanishes before \(\tau\) has probability tending to zero by the positive-risk assumption.
Gaussian limits and variance estimation
Theorem (Nelson-Aalen limit). Under the preceding assumptions and a martingale Lindeberg condition, \[ \sqrt n\{\widehat\Lambda-\Lambda\} \rightsquigarrow W\circ V \quad\text{in }D[0,\tau], \qquad V(t)=\int_0^t\frac{\lambda(u)}{y(u)}\,\mathrm{d}u, \] where \(W\) is standard Brownian motion. A consistent variance estimator for \(\widehat\Lambda(t)\) is \[ \widehat V_\Lambda(t) =\int_0^t\frac{1}{Y(u)^2}\,\mathrm{d}N(u) =\sum_{t_j\le t}\frac{d_j}{r_j^2}. \]
Proof. The scaled martingale has predictable variation \[ n\int_0^t\frac{\lambda(u)}{Y(u)}\,\mathrm{d}u \xrightarrow{P}V(t). \] Its jumps are bounded by \(\sqrt n/Y(u)=O_P(n^{-1/2})\), so the martingale functional central limit theorem applies. Replacing the compensator \(Y\lambda\,\mathrm{d}u\) by \(\mathrm{d}N\) gives \(\widehat V_\Lambda\); their difference is a martingale integral of smaller order.
Theorem (Greenwood formula). For the Kaplan-Meier estimator, \[ \widehat{\operatorname{Var}}\{\widehat S(t)\} =\widehat S(t)^2 \sum_{t_j\le t} \frac{d_j}{r_j(r_j-d_j)}. \] When ties are asymptotically negligible, \[ \sqrt n\{\widehat S(t)-S(t)\} =-S(t)\sqrt n\{\widehat\Lambda(t)-\Lambda(t)\}+o_P(1). \]
Proof. Conditional on each risk set, the estimated hazard jump \(d_j/r_j\) has binomial variance approximately \(q_j(1-q_j)/r_j\). The derivative of \(\log(1-q_j)\) is \(-1/(1-q_j)\), so its delta-method contribution is \[ \frac{q_j}{r_j(1-q_j)} \approx\frac{d_j}{r_j(r_j-d_j)}. \] Orthogonality of martingale increments permits summation; the exponential delta method returns to the survival scale. The functional expansion follows from Hadamard differentiability of \(\Lambda\mapsto e^{-\Lambda}\) in the continuous-hazard case.
Pointwise intervals and simultaneous bands
A normal interval on the raw survival scale can leave \([0,1]\). The complementary log-log transform \[ g(s)=\log\{-\log s\} \] maps \((0,1)\) to \(\mathbb R\) and respects cumulative-hazard geometry. If \(\widehat\sigma^2(t)\) is Greenwood's variance, then \[ \widehat{\operatorname{se}}\{g(\widehat S(t))\} =\frac{\widehat\sigma(t)} {|\widehat S(t)\log\widehat S(t)|}. \] Transforming a normal interval for \(g(S(t))\) back gives a pointwise interval inside \([0,1]\). Simultaneous Hall-Wellner or equal-precision bands instead use the functional Gaussian limit and a supremum critical value; replacing a pointwise \(z\)-quantile by the same number does not produce simultaneous coverage.