Second-order properties of an extrapolated bootstrap without replacement under weak assumptions

Abstract

This paper shows that a straightforward extrapolation of the bootstrap distribution obtained by resampling without replacement, as considered by Politis and Romano, leads to second-order correct confidence intervals, provided that the resampling size is chosen adequately. We assume only that the statistic of interest T_n, suitably renormalized by a regular sequence, is asymptotically pivotal and admits an Edgeworth expansion on some differentiable functions. The results are extended to a corrected version of the moving-block bootstrap without replacement introduced by Künsch for strong-mixing random fields. Moreover, we show that the generalized jackknife or the Richardson extrapolation of such bootstrap distributions, as considered by Bickel and Yahav, leads to better approximations.