Nelson–Aalen Estimator

The Nelson–Aalen estimator is a nonparametric estimator which may be used to estimate the cumulative hazard rate function from censored survival data (see Survival Distributions and Their Characteristics). Since no distributional assumptions are needed, one important use of the estimator is to check graphically the fit of parametric models, and this is the reason why it was originally introduced by Nelson [10, 11]. Independently of Nelson, Altshuler [2] derived the same estimator in the context of competing risks animal experiments. Later, by adopting a counting process formulation, Aalen [1] extended its use beyond the survival data and competing risks setups, and studied its small and large sample properties using martingale methods. The estimator is nowadays denoted the Nelson–Aalen estimator, although other names (the Nelson estimator, the Altshuler estimator, the Aalen–Nelson estimator, the empirical cumulative hazard estimator) are sometimes used as well. Below we present a number of situations where the Nelson–Aalen estimator may be applied and exemplify its use in one particular case. Furthermore, we indicate how counting processes provide a framework which allows for a unified treatment of all these diverse situations, and we summarize the most important properties of the Nelson–Aalen estimator. A detailed account is given in [3, Section IV.1].