Jackknife and bootstrap variance of a sample second moment (${\hat{v}}_J=\frac{1}{n(n-1)}{\sum }_i(x_i^2-\overline{x^2}{)}^2$)

For independent identically distributed observations with finite fourth moment, the sample mean of their squares has sampling variance $\mathrm{Var}(X^2)/n$. Its jackknife variance estimator is exactly the displayed unbiased estimator, since its leave-one-out values differ from their average by $-(x_i^2-\overline{x^2})/(n-1)$. The conditional bootstrap variance of a sample mean is ${\hat{v}}_B=n^{-2}{\sum }_i(x_i^2-\overline{x^2}{)}^2=(n-1){\hat{v}}_J/n$. Thus the two methods have a known finite-sample scaling difference, despite sharing the same large-sample target.