We first consider the random censorship model of survival analysis. The pairs of positive random variables ($X sb{i},Y sb{i}$), i = 1,$ ...$,n, are independent and identically distributed, with distribution functions F(t) = P($X sb{i} leq t$) and G(t) = P($Y sb{i} leq t$) and the Y's are independent of the X's. We observe only ($T sb{i}, delta sb{i}$), i = 1,$ ...$,n, where $T sb{i}$ = min($X sb{i},Y sb{i}$) and $ delta sb{i}$ = I($X sb{i} leq Y sb{i}$). The X's represent survival times, the Y's represent censoring times. Efron (1981) proposed two bootstrap methods for the random censorship model and showed that they are distributionally the same. Akritas (1986) established the weak convergence of the bootstrapped Kaplan-Meier estimator of F when bootstrapping is done by this method. Let us now consider bootstrapping more closely. Suppose that we wish to estimate the variance of F(t). If we knew the Y's then we would condition on them by the ancillarity principle, since the distribution of the Y's does not depend on F. That is, we would want to estimate Var$ {$F(t)$ vert Y sb1, ...,Y sb{n} }$. Unfortunately, in the random censorship model we do not see all the Y's. If $ delta sb{i}$ = 0 we see the exact value of $Y sb{i}$, but if $ delta sb{i}$ = 1 we know only that $Y sb{i} > T sb{i}$. Let us denote this information on the Y's by ${ cal C}$. Thus, what we want to estimate is Var$ {$F(t)$ vert{ cal C} }$. Efron's scheme is appropriate for estimating the unconditional variance. We propose a new bootstrap method which provides an estimate of Var$ {$F(t)$ vert{ cal C} }$.
