On Bootstrap Procedures for Second-order Accurate Confidence Limits in Parametric Models

This paper concerns the use of simulation procedures to construct secondorder accurate con dence limits having coverage error of order O(n ). An explicit formula for the analytical adjustment required in Efron's (1987) BCa percentile method is derived, automatic percentile methods that do not require analytical adjustments are proposed, and variance-stabilizing transformations designed to improve the performance of the bootstrap-t method are given. The automatic percentile methods and variance-stabilizing transformations involve a least favorable family construction of Stein (1956), which is related to orthogonal parameters. Connections with approximate limits obtained using pro le likelihood methods are also discussed.