# Standard error of the $n^n$ bootstrap means

I need to show that the standard error of the $$n^n$$ bootstrap means is $$SE^*(\bar{Y^*}) = \frac{S\sqrt{n-1}}{n}$$, where $$\bar{Y^*}$$ is the sample mean of a randomly drawn bootstrap sample, and $$S^2 = \frac{1}{n-1}\sum_{i=1}^{n}(Y_i - \bar{Y})^2$$. I know that $$SE^*(\bar{Y^*}) = \sqrt{\frac{\sum_{b=1}^{n^n}(\bar{Y}^*_b - \bar{Y})^2}{n^n}}$$, and have a hint that says I "should exploit the fact that the mean is a linear function of the observations."

Note that the "$$n^n$$ bootstrap means" is simply the bootstrap procedure in which all $$n^n$$ possible bootstrap samples are enumerated.

Thank you for any help!

In the the original sample of $$n$$ items $$Y_i$$ you have $$\sum Y_i = n \bar Y$$ and $$\sum (Y_i-\bar Y)^2 = (n-1) S^2$$.
In effect the original sample is the population for the bootstrap samples and you can treat each bootstrap sample as $$n$$ i.i.d. samples from a population with mean $$\bar Y$$ and variance $$\frac{n-1}{n}S^2$$ (as this is now being treated as a population you need to uncorrect for sample bias). So the bootstrap sums have mean $$n\bar Y$$ and second moment about $$n\bar Y$$ of $$(n-1)S^2$$
This implies that the bootstrap means have $$E[\bar Y_b^*] = \bar Y$$ and second moment about $$\bar Y$$ of $$E[(\bar Y_b^*- \bar Y)^2] = \frac{n-1}{n^2}S^2$$ i.e. with $$\sqrt{E[(\bar Y_b^*- \bar Y)^2]} = \sqrt{ \frac{n-1}{n^2}S^2} = \frac{\sqrt{n-1}}{n}S$$.
Since you have all $$n^n$$ possible bootstrap samples from the original sample, these expectations are realised, so $$\frac{\sum_{b=1}^{n^n}(\bar{Y}^*_b - \bar{Y})^2}{n^n} = \frac{n-1}{n^2}S^2$$ and $$SE^*(\bar{Y^*}) = \sqrt{\frac{\sum_{b=1}^{n^n}(\bar{Y}^*_b - \bar{Y})^2}{n^n}} = \frac{\sqrt{n-1}}{n}S$$
• What exactly do you mean when you say "the bootstrap sums have mean $n\bar{Y}$ and second moment about $n\bar{Y}$ of $(n-1)S^2$? Are you saying that $E(\sum_i\bar{Y^*_i}) = n\bar{Y}$ and $E((n\bar{Y})^2) = (n-1)S^2$? – Jake Dec 5 '19 at 19:05
• @Jake Not quite and I am not sure about your notation. A bootstrap sample is of $n$ items with replacement from the original sample, and the bootstrap sum is the sum of all $n$. So perhaps $E(\sum_i Y^*_i) = n\bar{Y}$ and $E(((\sum_i Y^*_i)-n\bar{Y})^2) = (n-1)S^2$. Then a bootstrap mean is $\bar{Y}^* = \frac1n \sum_i Y^*_i$ – Henry Dec 5 '19 at 19:20