On the relative performance of the block bootstrap for dependent data

In the independent setting, both Efron's bootstrap and `'empirical Edgeworth expansion'' (E.E-expansion) give second-order accurate approximations to distributions of standardized and studentized statistics in the smooth function model. As a result, Efron's bootstrap was often regarded as roughly equivalent to the one-term E.E-expansion. However, a more detailed analysis shows that Efron's bootstrap outperforms the E.E-expansion in terms of loss functions by Bhattacharya and Qumsiyeh (1989) and in terms of probabilities for large deviations by Hall (1990) and Jing et al (1994). In this paper, we shall study the performances of the block bootstrap and the E.E-expansion for the weakly dependent data. It turns out that similar properties hold: both perform equally well at the center of the distribution but the block bootstrap provides accurate approximations even in the tails of the distributions. The study is focused on the simple case of standardized and studentized sample mean, but the conclusions can be easily extended to the smooth function of multivariate means