On Bootstrap Coverage Probability with Dependent Data

This paper establishes the optimal bootstrap block lengths for coverage probabilities when the bootstrap is applied to covariance stationary ergodic dependent data. It is shown that the block lengths that minimize the error in coverage probabilities of one- and two-sided block bootstrap confidence intervals of normalized and studentized smooth functions of sample averages are proportional to $n^{1/4}$. The minimum error rates in coverage probabilities of one- and two-sided block bootstrap confidence intervals are of order O($n^{-3/2}$) and O($n^{-5/4}$), respectively, for normalized and studentized statistics. This constitutes a refinement over the asymptotic confidence intervals.