Strong Uniform Consistency for Nonparametric Survival Curve Estimators from Randomly Censored Data

Abstract

Let $X_1, \cdots, X_n$ be i.i.d. $P(X > u) = F(u)$ and $Y_1, \cdots, Y_n$ be i.i.d. $P(Y > u) = G(u)$, where both $F$ and $G$ are unknown continuous distributions. For $i = 1, \cdots, n$ set $\delta_i = 1$ if $X_i \leq Y_i$ and 0 if $X_i > Y_i$ and $Z_i = \min \{X_i, Y_i\}$. One way to estimate $F$ from the observations $(Z_i, \delta_i) i = 1, \cdots, n$ is by means of the product limit (PL) estimator, $F^\ast_n$ (Kaplan-Meier, [8]). In this paper it is shown that $F^\ast_n$ is uniformly almost sure consistent with rate $O(\sqrt{\log n} / \sqrt n)$, that is $P(\sup_{0 \leq u \leq T}|F^\ast_n(u) - F(u)| = O(\sqrt{\log n/n})) = 1.$ Assuming that $F$ is distributed according to a Dirichlet process (Ferguson, [3]) with parameter $\alpha$, Susarla and Van Ryzin ([11]) obtained the Bayes estimator $F^\alpha_n$ of $F$. In the present paper a similar result is established for the Bayes estimator, namely: $P(\sup_{0 \leq u \leq T}| F^\alpha_n(u) - F(u)| = O(\sqrt{(\log n)^{1 + \gamma}} / \sqrt n)) = 1 \quad (\gamma > 0).$