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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 bootstraps and without replacement works. 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 doesn't.

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