Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses

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.