In the non-parametric bootstrap, the configuration \(B\) is regarded as a list of \(n\) tables in order of occupation. Each non-parametric bootstrap sample is a sequence of \(n\) tables drawn with replacement from the empirical distribution of tables. Although \(\hat{\alpha}\left(B^{*}ight) \leq \hat{\alpha}(B)\) for every bootstrap sample, show that

\[
\begin{aligned}
E\left(\# B-\# B^{*}ight) & =-\alpha \log \left(1-e^{-1}ight)+o(1) \\
E\left(\operatorname{var}\left(\# B^{*}ight) \mid Bight) & =-\alpha \log \left(1-e^{-1}ight)+\alpha \log \left(1-e^{-2}ight)+o(1)
\end{aligned}
\]

Hence deduce that the sample variance of bootstrap estimates satisfies

\[
E\left(\operatorname{var}\left(\hat{\alpha}\left(\# B^{*}ight)ight) \mid Bight) \simeq \alpha \text { const } / \log ^{2} n
\]

in order-of-magnitude agreement with Fisher's calculation. Show that the bootstrap constant is not the same as Fisher's constant in (11.7).