  The recent seminal work of Chernozhukov, Chetverikov and Kato has shown that
bootstrap approximation for the maximum of a sum of independent random vectors
is justified even when the dimension is much larger than the sample size. In
this context, numerical experiments suggest that third-moment match bootstrap
approximations would outperform normal approximation even without
studentization, but the existing theoretical results cannot explain this
phenomenon. In this paper, we first show that Edgeworth expansion, if
justified, can give an explanation for this phenomenon. Second, we obtain valid
Edgeworth expansions in the high-dimensional setting when the random vectors
have Stein kernels. Finally, we prove the second-order accuracy of a double
wild bootstrap method in this setting. As a byproduct, we find an interesting
blessing of dimensionality phenomenon: The single third-moment match wild
bootstrap is already second-order accurate in high-dimensions if the covariance
matrix has identical diagonal entries and bounded eigenvalues.
