Stute and Wang (1994) considered the problem of estimating the integral SΦ = ∫ ΦdF, based on a possibly censored sample from a distribution F, where Φ is an F-integrable function. They proposed a Kaplan-Meier integral ?Φn to approximate SΦ and derived an explicit formula for the delete-1 jackknife estimate ?Φn(1). ?Φn(1) differs from ?Φn only when the largest observation, X(n), is not censored (δ(n) = 1) and next-to-the-largest observation, X(n-1), is censored (δ(n-1) = 0). In this note, it will be pointed out that when X(n) is censored (δ(n) =0) ?Φn(1) is based on a defective distribution, and therefore ?Φn(1) can badly underestimate SΦ. We derive an explicit formula for the delete-2 jackknife estimate ?Φn(2). However, on comparing the expressions of ?Φn(1) and ?Φn(2), their difference is negligible. To improve the performance of ?Φn(1) and ?Φn(2), we propose a modified estimator S?Φn according to Efron (1980). Simulation results demonstrate that S?Φn is much less biased than ?Φn(1) and ?Φn(2). ? 1998 Elsevier Science B.V. All rights reserved.
Relation:
Statistics and Probability Letters Volume 40, Issue 4, 15 November 1998, Pages 353-361
