Statistica Sinica 7(1997), 1-31

RESAMPLING FEWER THAN n OBSERVATIONS:

GAINS, LOSSES, AND REMEDIES FOR LOSSES

P. J. Bickel, F. Götze and W. R. van Zwet

University of California, Berkeley,

University of Bielefeld and University of Leiden

Abstract: We discuss a number of resampling schemes in which m=o(n) observations are resampled. We review nonparametric bootstrap failure and give results old and new on how the m out of n with replacement and without replacement bootstraps work. We extend work of Bickel and Yahav (1988) to show that m out of n bootstraps can be made second order correct, if the usual nonparametric bootstrap is correct and study how these extrapolation techniques work when the nonparametric bootstrap does not.

Key words and phrases: Asymptotic, bootstrap, nonparametric, parametric, testing.