Prepivoting by weighted bootstrap iteration

Prepivoting by conventional bootstrap iteration is known to yield a progressively more accurate pivot in certain problems, and has important application in the construction of confidence limits and estimation of null distributions.We investigate the theoretical effects of weighted bootstrap iteration on prepivoting and show that each weighted bootstrap iteration, with weights chosen carefully but empirically, is asymptotically equivalent to two consecutive conventional bootstrap iterations. In terms of reducing the order of error, prepivoting can therefore be carried out much more efficiently if based on weighted bootstrap iterations. This is shown for a variety of problem settings, including the smooth function model, M-estimation and the regression context. A numerical illustration is provided, demonstrating the potential practical usefulness of weighted prepivoting.