Nelson–Aalen estimator

${\displaystyle \tilde{H}(t)=\sum _{t_i\le t}\frac{d_i}{n_i},}$

with ${\displaystyle d_i}$ the number of events at time ${\displaystyle t_i}$ and ${\displaystyle n_i}$ the total individuals at risk at ${\displaystyle t_i}$.

The curvature of the Nelson–Aalen estimator gives an idea of the hazard rate shape. A concave shape is an indicator for infant mortality while a convex shape indicates wear out mortality.

It can be used for example when testing the homogeneity of Poisson processes.

It was constructed by Wayne Nelson and Odd Aalen. The Nelson-Aalen estimator is directly related to the Kaplan-Meier estimator and both maximize the empirical likelihood.