# Subsampling vs. m out of n bootstrap

My understanding is the distinction between subsampling and the m out of n bootstrap is that subsampling draws without replacement but the m out of n bootstrap does not.

If we are not in a situation with m = n, is there a reason to prefer the m out of n bootstrap? I am struggling to think of an intuitive reason why subsampling would not dominate, but I realize there may be some math justifying m out of n.

• You may want to look at jackknife (en.wikipedia.org/wiki/Jackknife_resampling) for similar intuitions. Sometimes, if you have a truly large sample you need to depend on m out of n sampling. Jun 28, 2021 at 16:15

Theoretically, when $$m \ll n$$, it does not matter if you sample with or without replacement (if you sample with replacement and $$m \ll n$$ the probability of drawing the same observation twice can be made arbitrarily small as $$\frac{m}{n} \to 0$$).
In practice, the two procedures usually provide very similar results. The key assumption that makes subsampling valid is that $$m \to \infty$$ and $$\frac{m}{n} \to 0$$ as $$n \to \infty$$. (To be precise if $$\theta_n$$ is your test statistic and $$\theta$$ is the true value and if $$\tau_n (\theta_n - \theta)$$ converges to a limit law, then $$m \to \infty$$, $$\frac{m}{n} \to 0$$ and $$\frac{\tau_m}{\tau_n} \to 0$$ have to hold if $$n \to \infty$$).