NelsonAalenestimator

The Nelson-Aalen estimator is a nonparametric estimator of the cumulative hazard function in survival analysis for time-to-event data with right censoring. It is named after Edward Nelson and Odd Aalen and provides a way to summarize how the hazard accumulates over time without assuming a specific parametric form.

Let t1 < t2 < ... denote observed event times, dN(t) the number of events at time t, and Y(t) the number at risk just before t. The Nelson-Aalen estimator of the cumulative hazard Λ(t) up to time t is

Λ_hat(t) = sum over event times s ≤ t of dN(s) / Y(s).

The estimator is a right-continuous step function with jumps at observed event times.

Properties and interpretation: The estimator is nonparametric and consistent under independent censoring. It arises from counting-process martingale theory and has asymptotic normality. An approximate variance can be estimated by

Var(Λ_hat(t)) ≈ sum_{s ≤ t} dN(s) / Y(s)^2,

which yields standard errors and confidence bands.

Relation to survival function and regression: The survival function is related to the cumulative hazard by S(t) = exp(-Λ(t)) in the continuous-time hazard model, so S_hat(t) ≈ exp(-Λ_hat(t)). In Cox proportional hazards models, the Nelson-Aalen estimator can be used to estimate the cumulative baseline hazard Λ0(t); it is related to the Breslow-type estimator used in this context.

Applications and implementation: The Nelson-Aalen estimator is widely used in clinical trials and reliability studies for analyzing time-to-event data with censoring. It is implemented in standard survival-analysis software and accompanies other nonparametric methods for hazard and survival estimation.