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
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
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
Applications and implementation: The Nelson-Aalen estimator is widely used in clinical trials and reliability studies for