A Review of Bootstrap Confidence Intervals

A survey of bootstrap procedures for constructing confidence regions is given. In particular, several distinct bootstrap methods are considered, with emphasis on the mathematical correctness of these procedures. The percentile, bias-corrected percentile and accelerated bias-corrected percentile methods, developed by Efron, are reviewed in both parametric and nonparametric situations. A procedure related to the accelerated bias-corrected method, which avoids explicit calculation of the analytical corrections required in Efron's method, is also introduced. In the context of a functional approach for the construction of confidence regions, the bootstrap is motivated as a method to estimate the distributions of approximate pivots. Finally, iterative bootstrap methods are discussed as means to improve coverage accuracy.