  The block bootstrap confidence interval based on dependent data can
outperform the computationally more convenient normal approximation only with
non-trivial Studentization which, in the case of complicated statistics, calls
for highly specialist treatment. We propose two different approaches to
improving the accuracy of the block bootstrap confidence interval under very
general conditions. The first calibrates the coverage level by iterating the
block bootstrap. The second calculates Studentizing factors directly from block
bootstrap series and requires no non-trivial analytic treatment. Both
approaches involve two nested levels of block bootstrap resampling and yield
high-order accuracy with simple tuning of block lengths at the two resampling
levels. A simulation study is reported to provide empirical support for our
theory.
