Pitfalls in Constructing Bootstrap Confidence Intervals for Asymptotically Pivotal Statistics

The conventional Edgeworth expansion view of bootstrap confidence intervals suggests that for the bootstrap to exceed the accuracy of the normal approximation one must bootstrap asymptotically pivotal statistics. This paper questions the basic premise of the asymptotic theory used to rationalize the higher-order accuracy of bootstrap intervals for asymptotically pivotal statistics. In finite samples, these statistics often are not even approximately pivotal. As a result, Edgeworth expansion arguments for pivotal statistics do not apply, and the only way to compare the accuracy of alternative intervals is by simulation. The paper documents that percentile-t intervals based on asymptotic pivots tend to behave erratically in small samples and may be much less accurate than bootstrap intervals based on nonpivotal statistics. It is also shown that bootstrap intervals can be very accurate in the absence of asymptotic refinements, and that there are huge differences in coverage accuracy among asymptotically equivalent intervals that cannot be explained by Edgeworth expansion arguments.