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

Abstract

Let X1, Xn be i.i.d. P(X > u) = F(u) and Y1, Yn be i.i.d. P(Y > u) = G(u) , where both F and G are unknown continuous distributions. For i = 1, n set δi = 1 if Xi Yi and 0 if Xi > Yi and Zi = min Xi, Yi. One way to estimate F from the observations (Zi, δi) i = 1, n is by means of the product limit (PL) estimator, F n (Kaplan-Meier, 8). In this paper it is shown that F n is uniformly almost sure consistent with rate O(sqrtlog n sqrt n) , that is P(sup0 leq u leq TF astn(u) - F(u) = O(sqrtlog n/n)) = 1. Assuming that F is distributed according to a Dirichlet process (Ferguson, 3) with parameter α, Susarla and Van Ryzin (11) obtained the Bayes estimator Fα n of F. In the present paper a similar result is established for the Bayes estimator, namely: P(sup0 leq u leq T F alphan(u) - F(u) = O(sqrt(log n) 1 + gamma sqrt n)) = 1 quad (gamma > 0).$