Nelson–Aalen estimator

From HandWiki

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

[math]\displaystyle{ \tilde{H}(t)=\sum_{t_i\leq t}\frac{d_i}{n_i}, }[/math]

with [math]\displaystyle{ d_i }[/math] the number of events at [math]\displaystyle{ t_i }[/math] and [math]\displaystyle{ n_i }[/math] the total individuals at risk at [math]\displaystyle{ t_i }[/math].[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]


  1. "Kaplan–Meier and Nelson–Aalen Estimators". 21 September 2008. 
  2. "Kaplan–Meier Survival Estimates". 
  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. doi:10.1016/j.gloplacha.2010.03.006. Bibcode2010GPC....72...55K. 
  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. 
  7. Zhou, M. (2015). Empirical Likelihood Method in Survival Analysis (1st ed.). Chapman and Hall/CRC.,

Further reading

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