Upper bounds on the true coverage of bootstrap percentile type confidence intervals

Since its inception, a major use of the bootstrap methodology has been in t he construction of approximate nonparametric confidence intervals. As evide nced by many spirited discussions over the past few years, the best way of constructing these intervals has not been resolved. In particular, empirica l studies have shown that many of these intervals have disappointing finite sample coverage probabilities. The purpose of this article is to show that intervals based on percentiles of the bootstrap distribution have bounds o n their finite sample coverage probabilities. Depending on the functional o f interest and the distribution of the data, these bounds can be quite low. We argue that these bounds are valid even for methods that are asymptotica lly second-order accurate. These results are useful to researchers who are contemplating using this type of confidence interval when the sample size i s small. These bounds are computed for several examples including the momen ts and quantiles of several distributions.