Nelson–Aalen estimator

{{Short description|Nonparametric estimate of cumulative hazard}}

The Nelson–Aalen estimator is a non-parametric estimator of the cumulative hazard rate function in case of censored data or incomplete data.{{cite web |url=http://tkchen.wordpress.com/2008/09/21/kaplan-meier-and-nelson-aalen-estimators/|title=Kaplan–Meier and Nelson–Aalen Estimators|date=21 September 2008 }} 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

:$\backslash tilde\{H\}(t)=\backslash sum\_\{t\_i\backslash leq\; t\}\backslash frac\{d\_i\}\{n\_i\},$

with $d\_i$ the number of events at time $t\_i$ and $n\_i$ the total individuals at risk at $t\_i$.{{cite web |url=http://www.statsdirect.com/help/survival_analysis/kaplan.htm|title=Kaplan–Meier Survival Estimates}}

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.{{cite journal |first1=Jan |last1=Kysely |first2=Jan |last2=Picek |first3=Romana |last3=Beranova |year=2010 |title=Estimating extremes in climate change simulations using the peaks-over-threshold method with a non-stationary threshold |journal=Global and Planetary Change |volume=72 |issue=1–2 |pages=55–68 |doi=10.1016/j.gloplacha.2010.03.006 |bibcode=2010GPC....72...55K }}

It was constructed by Wayne Nelson and Odd Aalen.{{cite journal |last1=Nelson |first1=W. |title=Hazard plotting for incomplete failure data. |journal=Journal of Quality Technology |date=1969 |volume=1 |pages=27–52 |doi=10.1080/00224065.1969.11980344}}{{cite journal |last1=Nelson |first1=W. |title=Theory and applications of hazard plotting for censored failure data |journal=Technometrics |date=1972 |volume=14 |issue=4 |pages=945–965 |doi=10.1080/00401706.1972.10488991}}{{cite journal |last1=Aalen |first1=Odd |title=Nonparametric inference for a family of counting processes|journal=Annals of Statistics |date=1978 |volume=6 |issue=4 |pages=701–726 |jstor=2958850 |url=https://projecteuclid.org/download/pdf_1/euclid.aos/1176344247 |doi=10.1214/aos/1176344247|doi-access=free |url-access=subscription }}

The Nelson-Aalen estimator is directly related to the Kaplan-Meier estimator and both maximize the empirical likelihood.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