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.


1 Answer 1


You are correct.

The bootstrap samples with replacement while subsampling samples without replacement.

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 $n \to \infty$).


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