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.