On Bootstrap Iteration for Coverage Correction in Confidence Intervals

Abstract

Abstract Iterated bootstrap procedures may be used to reduce error in many statistical problems. We discuss their use in constructing confidence intervals with accurate coverage and show that bootstrap coverage correction produces improvements in coverage accuracy of order n −1/2 in one-sided intervals, but of order n −1 in two-sided intervals. Explicit formulas are provided for the dominant term in coverage error after iteration in each case. These results are used to compare various iterated bootstrap intervals and to assess the effect of bootstrap iteration on other indicators of interval performance, such as position of critical points and length of interval. We show that, for one-sided intervals, the coverage-correction algorithm yields critical points that are second-order correct. This is not the case for two-sided intervals, where second-order correctness is not crucial in obtaining high-order coverage accuracy. We also show that the asymptotic mean increase in length between the original and coverage-corrected intervals is proportional to the coverage error of the original interval. The arguments are illustrated via several examples, including the application of the bootstrap coverage-correction algorithm to reduce coverage error of Bartlett-corrected likelihood-based confidence regions from order n −2 to order n −3.