A Note on the Uniform Consistency of the Kaplan-Meier Estimator

Abstract

Let $\{X_n, n \geq 1\}$ be i.i.d with $P(X_i \leq u) = F(u)$ and $\{U_n, n \geq 1\}$ be i.i.d. with $P(U_i \leq u) = G(u). \hat{F}_n(t)$ is the Kaplan-Meier estimator based on the censored data $(\tilde{X}_i = X_i \wedge U_i, \delta_i = 1_{(X_i \leq U_i)}, 1 \leq i \leq n)$. In this note, it is shown that for $T_n = \max_{1 \leq i \leq n} \tilde{X}_i$, $\mathrm{pr}-\lim_{n \rightarrow \infty} \sup_{t \leq T_n} |\hat{F}_n(t) - F(t)| = 0.$ Hence, the largest interval on which the Kaplan-Meier estimator is uniformly consistent is found.