Kaplan-Meier Estimator

From Open Risk Manual

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 i in continuous time t is a Random Variable R^i(t) taking values in the state space S (We assume a finite state space S ={0, D}), where 0 is the live (healthy / performing) state and D is the dead (non-performing) state.

Denote t_1 < t_2 < \dots t_n the times at which entities transition from state 0 to state D and let d_j the cumulative count of such transitions at time t_j. Then the estimator is given by the expression:


\hat{S}(t) = \prod_{t_j \le t} (1 - \frac{d_j}{r_j})

where r_j is the number of entities that are alive prior to time t_j.

The Kaplan-Meier hazard rate estimator is simply


\hat{\lambda}(t_j) = \frac{d_j}{r_j})

The Nelson-Aalen estimator for the cumulative hazard is


\hat{\lambda}(t) = \sum_{t_j \le t} \frac{d_j}{r_j}

Variance

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


\hat{\sigma}^2(t) = (\hat{S}(t))^2 \sum_{t_j \le t}  \frac{d_j}{r_j (r_j - d_j)}

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:


\hat{S}(t) = \frac{1}{N} \sum_{j=1}^{N} 1_{t_j > t}

See Also

References

  1. 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.

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