AN IMPROVEMENT OF THE JACKKNIFE DISTRIBUTION FUNCTION ESTIMATOR

In a recent paper, C. F. J. Wu showed that the jackknife estimator of a distribution function has optimal convergence rate O(n-1/2), where n denotes the sample size. This rate is achieved by retaining O(n) data values from the original sample during the jackknife algorithm. Wu's result is particularly important since it permits a direct comparison of jackknife and bootstrap methods for distribution estimation. In the present paper we show that a very simple, nonempirical modification of the jackknife estimator improves the convergence rate from O(n-1/2) to O(n-5/6), and that this rate may be achieved by retaining only O(n2/3) data values from the original sample. Our technique consists of mixing the jackknife distribution estimator with the standard normal distribution in an appropriate proportion. The convergence rate of O(n-5/6) makes the jackknife significantly more competitive with the bootstrap, which enjoys a convergence rate of O(n-1) in this particular problem.