A Combinatoric Approach to the Kaplan-Meier Estimator

Abstract

The paper considers the Kaplan-Meier estimator $F^{\mathrm{KM}}_n$ from a combinatoric viewpoint. Under the assumption that the estimated distribution $F$ and the censoring distribution $G$ are continuous, the combinatoric results are used to show that $\int |\theta(z)| dF^{\mathrm{KM}}_n(z)$ has expectation not larger than $\int |\theta(z)| dF(z)$ for any sample size $n$. This result is then coupled with Chebychev's inequality to demonstrate the weak convergence of the former integral to the latter if the latter is finite, if $F$ and $G$ are strictly less than 1 on $\mathscr{R}$ and if $\theta$ is continuous.