Bootstrapping the Kaplan—Meier estimator

Randomly censored data consist of iid pairs of observations (X_i, δ_i), i = 1, …, n; if δ_i= 0, X_idenotes a censored observation, and if δ_i= 1, X_idenotes an exact “survival” time, which is the variable of interest. For estimating the distribution F of the survival times, the product-limit estimator proposed by Kaplan and Meier (1958) has been studied extensively and it has been shown to enjoy a number of optimality properties (Wellner 1982). See Gill (1980, chaps. 1–4; 1983) for a modern treatment and for references. With censored data, bootstrapping can be carried out using two different resampling plans introduced by Efron (1981) and Reid (1981), respectively. With Efron’s plan one takes a random sample with replacement from (X₁), …, (X_n, δ_n), whereas with Reid’s plan one takes a random sample from the Kaplan—Meier estimator. The purpose of this article is to study the asymptotic behavior of the bootstrapped Kaplan—Meier estimator with both resampling plans. The approach adopted uses the theory of martingales for point processes. This extends our understanding of the asymptotic behavior of bootstrapped estimated distribution functions to the situation in which random censoring occurs. The uncensored case has been studied by Bickel and Freedman (1981) and Shorack (1982). The new result is used to obtain bootstrap confidence bands for a survival distribution under random censoring using Efron’s approach. These bands are the only available ones that are valid even when the survival distribution has a discrete component. Further, it is demonstrated that Reid’s proposal for bootstrapping does not produce asymptotically correct confidence bands. Results from simulation studies assessing the finite sample performance of the bootstrap confidence bands are included.