Jackknife estimation of the bootstrap acceleration constant

Abstract

Efron (1987) describes a new procedure for obtaining better bootstrap confidence intervals, involving the “acceleration” constant, a. He estimates a using the infinitesimal jackknife (a finite sample version of the first order influence function) of the estimator θ for a parameter θ. In this paper, the ordinary positive and negative jackknife estimates of the influence function, which involve adding or deleting single observations, are used to obtain alternative estimators of a. Aspects of the performance of the resulting confidence intervals are compared in a simulation study for the variance and the correlation coefficient. One conclusion is that the accelerated bootstrap does result in improved coverage. However, it is neither uniformly so in all cases nor enough to correct the entire shorfall from the nominal. Secondly, the ordinary jackknife is the preferred approach to the problem of estimating a. Additional simulations compare various versions of bootstrap t intervals with a modified jackknife and normal theory intervals. The studentized intervals are significantly better quantities to bootstrap, yielding much closer agreement with nominal coverage.