# Kaplan-Meier Estimator

## Definition

The **Kaplan-Meier** estimator is a nonparametric estimator^{[1]} 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.

## Estimator

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

## Variance

The variance of the Kaplan-Meier estimator is given by Greenwood's formula:

## No Censoring

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:

## See Also

## References

- ↑ 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.