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.

翻译：Chernozhukov、Chetverikov与Kato近期的开创性工作表明，即使维度远大于样本量，独立随机向量和的最大值的自助法逼近仍然有效。在此背景下，数值实验表明三阶矩匹配自助法逼近即使未经学生化处理也能优于正态逼近，但现有理论结果无法解释这一现象。本文首先证明：若Edgeworth展开成立，则可为此现象提供理论解释。其次，我们在随机向量具有Stein核的高维设定下获得了有效的Edgeworth展开。最后，我们在此设定下证明了双重野生自助法（double wild bootstrap）的二阶精度。作为推论，我们发现了一个有趣的维度福音现象：若协方差矩阵具有相同对角元且有界特征值，则单一的三阶矩匹配野生自助法在高维情形下已具有二阶精度。