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.