Nelson–Aalen estimator explained

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

\tilde{H}(t) = \sum _{t _{i\leq} t} (d _{i})/(n _{i}) ,

with

d _{i} the number of events at time t _{i} and n _{i} the total individuals at risk at 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]

Further reading

• Book: Jones . Andrew M. . Nigel . Rice . Teresa Bago . D'Uva . Silvia . Balia . Duration Data . Applied Health Economics . London . Routledge . 2013 . 978-0-415-67682-3 . 139–181 . https://books.google.com/books?id=7tdcCol9mNEC&pg=PA141 .

Notes and References

1. Web site: Kaplan–Meier and Nelson–Aalen Estimators. 21 September 2008 .
2. Web site: Kaplan–Meier Survival Estimates.
3. Jan . Kysely . Jan . Picek . Romana . Beranova . 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 . 10.1016/j.gloplacha.2010.03.006 . 2010GPC....72...55K .
4. Nelson . W. . Hazard plotting for incomplete failure data. . Journal of Quality Technology . 1969 . 1 . 27–52 . 10.1080/00224065.1969.11980344.
5. Nelson . W. . Theory and applications of hazard plotting for censored failure data . Technometrics . 1972 . 14 . 4 . 945–965 . 10.1080/00401706.1972.10488991.
6. Aalen . Odd . Nonparametric inference for a family of counting processes. Annals of Statistics . 1978 . 6 . 4 . 701–726 . 2958850 . 10.1214/aos/1176344247. free .
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