# 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 time ${\displaystyle t_{i}}$ 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]

## References

1. ^ "Kaplan–Meier and Nelson–Aalen Estimators". 21 September 2008.
2. ^
3. ^ Kysely, Jan; Picek, Jan; Beranova, Romana (2010). "Estimating extremes in climate change simulations using the peaks-over-threshold method with a non-stationary threshold". Global and Planetary Change. 72 (1–2): 55–68. Bibcode:2010GPC....72...55K. doi:10.1016/j.gloplacha.2010.03.006.
4. ^ Nelson, W. (1969). "Hazard plotting for incomplete failure data". Journal of Quality Technology. 1: 27–52. doi:10.1080/00224065.1969.11980344.
5. ^ Nelson, W. (1972). "Theory and applications of hazard plotting for censored failure data". Technometrics. 14 (4): 945–965. doi:10.1080/00401706.1972.10488991.
6. ^ Aalen, Odd (1978). "Nonparametric inference for a family of counting processes". Annals of Statistics. 6 (4): 701–726. doi:10.1214/aos/1176344247. JSTOR 2958850.
7. ^ Zhou, M. (2015). Empirical Likelihood Method in Survival Analysis (1st ed.). Chapman and Hall/CRC. https://doi.org/10.1201/b18598, https://books.google.com/books?id=9-b5CQAAQBAJ&dq=Does+the+Nelson%E2%80%93Aalen+estimator+construct+an+empirical+likelihood%3F&pg=PA7