Subsampling and the m-out-of-n Bootstrap

The ordinary bootstrap resamples $n$ observations with replacement from a dataset of size $n$, and usually gives a reliable picture of an estimator's sampling distribution. But it quietly breaks down in certain situations — estimating an extreme value (like a maximum), or estimating a quantity right at a boundary of its possible range — where resampling with replacement doesn't correctly mimic how the original data was generated, giving a confidence interval that's systematically wrong even as $n$ grows large.

Subsampling, and its close cousin the m-out-of-n bootstrap, fixes this by drawing samples of a smaller size $m < n$ — subsampling draws them without replacement (genuine subsets of the original data), while the m-out-of-n bootstrap draws with replacement but from a smaller sample size. Either version turns out to give a theoretically valid approximation to the sampling distribution in exactly the cases where the standard $n$-out-of-$n$ bootstrap fails, at the cost of needing to choose $m$ sensibly (commonly a rate like $m \propto n^{2/3}$).

A worked example

Estimating the worst single-day loss ever observed in a return series of $n = 1000$ days is exactly the kind of extreme-value problem where the ordinary bootstrap misbehaves: because the maximum can only repeat, not exceed, values already in the data, resampled maxima cluster too tightly and understate true tail uncertainty. Subsampling instead draws subsamples of, say, $m = 100$ days without replacement, computes the worst loss in each subsample, and studies how that quantity varies across many such subsamples — capturing genuine sampling variability in the tail estimate that the full-size bootstrap would have masked.