Exercise 4.3: Bootstrapping theory with… bootstrapping!

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Say we have a data set with \(n\) unique measurements.

a) Show that on average a fraction of \((1 - 1/n)^n\) of the measurements do not appear in a bootstrap sample. Note that for large \(n\), this is approximately \(1/\mathrm{e} \approx 1/2.7\), since \(\lim_{n\to\infty}(1 - 1/n)^n = 1/\mathrm{e}\). This part of the problem is optional and not graded.

b) Use a bootstrapping approach to demonstrate that this is indeed true. Hint: Think about a convenient “data set” to use for drawing samples. This is kind of fun; you’re investigating some theory behind bootstrapping with bootstrapping!