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.
@article{1176346582,
author = {Mauro, David},
title = {A Combinatoric Approach to the Kaplan-Meier Estimator},
journal = {Ann. Statist.},
volume = {13},
number = {1},
year = {1985},
pages = { 142-149},
language = {en},
url = {http://dml.mathdoc.fr/item/1176346582}
}
Mauro, David. A Combinatoric Approach to the Kaplan-Meier Estimator. Ann. Statist., Tome 13 (1985) no. 1, pp. 142-149. http://gdmltest.u-ga.fr/item/1176346582/