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
- [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 time [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]
References
- ↑ "Kaplan–Meier and Nelson–Aalen Estimators". 21 September 2008. http://tkchen.wordpress.com/2008/09/21/kaplan-meier-and-nelson-aalen-estimators/.
- ↑ "Kaplan–Meier Survival Estimates". http://www.statsdirect.com/help/survival_analysis/kaplan.htm.
- ↑ 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. Bibcode: 2010GPC....72...55K.
- ↑ Nelson, W. (1969). "Hazard plotting for incomplete failure data.". Journal of Quality Technology 1: 27–52. doi:10.1080/00224065.1969.11980344.
- ↑ Nelson, W. (1972). "Theory and applications of hazard plotting for censored failure data". Technometrics 14 (4): 945–965. doi:10.1080/00401706.1972.10488991.
- ↑ Aalen, Odd (1978). "Nonparametric inference for a family of counting processes". Annals of Statistics 6 (4): 701–726. doi:10.1214/aos/1176344247. https://projecteuclid.org/download/pdf_1/euclid.aos/1176344247.
- ↑ 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
Further reading
- Jones, Andrew M.; Rice, Nigel; D'Uva, Teresa Bago; Balia, Silvia (2013). "Duration Data". Applied Health Economics. London: Routledge. pp. 139–181. ISBN 978-0-415-67682-3. https://books.google.com/books?id=7tdcCol9mNEC&pg=PA141.
External links
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