The Kaplan-Meier estimator is a nonparametric estimator of the Survival Function from (possibly censored) data. It concerns the special case when the State Space of the stochastic system has only two states (Alive / Dead) and one of them is an absorbing state, that is, once the system reaches this state it never leaves.
The position in state space for an entity in continuous time is a Random Variable taking values in the state space S (We assume a finite state space ), where 0 is the live (healthy / performing) state and D is the dead (non-performing) state.
Denote the times at which entities transition from state 0 to state D and let the cumulative count of such transitions at time . Then the estimator is given by the expression:
where is the number of entities that are alive prior to time .
The Kaplan-Meier hazard rate estimator is simply
The Nelson-Aalen estimator for the cumulative hazard is
The variance of the Kaplan-Meier estimator is given by Greenwood's formula:
In the case of no censoring, the Kaplan-Meier estimator is equivalent to the empirical survival function. If the population involves N entities, this is given by:
- Kaplan, E. L. and Meier, P. (1958). Non-parametric estimation from incomplete observations. Journal of the American Statistical Association 53, 457–481 and 562– 563.