MOMENTS OF THE KAPLAN-MEIER ESTIMATOR

An exact moment expression of the Kaplan-Meier (KM) estimator has been obtained by Chen, Hollander and Langberg (1982) under the assumption of proportional hazard model. Without the proportional hazard assumption, the general moment formulas of the KM estimator are not available yet. Let FBAR(n) (t) be the KM estimator of the underlying survival function with right censored data. In this article we derive the von Mises expansion of the KM cumulative hazard function - In FBAR(n) (t) and then, we show that the moments of the KM estimator are power series of n-1 with functionals of subsurvival functions as coefficients. For the purpose of practical application, we obtain the second, the third and the fourth moments of the KM estimator with the accuracy of o(n-2). Numerical comparison of the approximate moments developed in this article with the exact moments under the proportional hazard model is presented.