Nelson–Aalen estimator

The Nelson–Aalen estimator is a non-parametric estimator of the cumulative hazard rate function in case of censored data or incomplete data.[1] It is used in survival theory, reliability engineering and life insurance to estimate the cumulative number of expected events. An "event" can be the failure of a non-repairable component, the death of a human being, or any occurrence for which the experimental unit remains in the "failed" state (e.g., death) from the point at which it changed on. The estimator is given by

${\displaystyle {\tilde {H}}(t)=\sum _{t_{i}\leq t}{\frac {d_{i}}{n_{i}}},}$

with ${\displaystyle d_{i}}$ the number of events at ${\displaystyle t_{i}}$ and ${\displaystyle n_{i}}$ the total individuals at risk at ${\displaystyle t_{i}}$.[2]

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.[3]

It was constructed by Wayne Nelson and Odd Aalen.[4][5][6] The Nelson-Aalen estimator is directly related to the Kaplan-Meier estimator and both maximize the empirical likelihood.[7]