Abstract

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 talk, we show that Edgeworth expansion, if justified, can give an explanation for this phenomenon. In particular, we derive an asymptotic expansion formula of the bootstrap coverage probability and show that the third-moment match wild bootstrap is second-order accurate in high-dimensions even without studentization when the covariance matrix has identical diagonal entries and bounded eigenvalues. In addition, we show the validity of the asymptotic expansion in the existence of so-called Stein kernels.