Why a bootstrap sample misses about a third of your data

Question

You draw a bootstrap sample: n points sampled with replacement from your n original observations. What fraction of the original points, on average, never appear in a given bootstrap sample, and why does this number matter?

Accepted Answer

The probability a point is missed. Focus on one specific observation. On each of the n independent draws, the chance it is not selected is 1 - 	frac{1}{n}. Across all n draws, P(	ext{never selected}) = \left(1 - \frac{1}{n}ight)^{n}. Take the limit. As n grows, this famously converges to \left(1 - \frac{1}{n}ight)^{n} 	o e^{-1} \approx 0.368. So about 36.8% of the original points are left out of any given bootstrap sample, and about 1 - e^{-1} \approx 63.2\% of the distinct points appear (the rest of the slots are duplicates). This is the origin of the "0.632" you see in resampling met…

Why a bootstrap sample misses about a third of your data

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