Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S[theta] = [integral operator] [theta] dF, based on a possibly censored sample from a distribution F, where [theta] is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S[theta] and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X(n), is not censored ([delta](n) = 1 and next-to-the-largest observation, X(n-1), is censored ([delta](n-1) = 0). In this note, it will pointed out that when X(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .