  We show that, when the double bootstrap is used to improve performance of
bootstrap methods for bias correction, techniques based on using a single
double-bootstrap sample for each single-bootstrap sample can be particularly
effective. In particular, they produce third-order accuracy for much less
computational expense than is required by conventional double-bootstrap
methods. However, this improved level of performance is not available for the
single double-bootstrap methods that have been suggested to construct
confidence intervals or distribution estimators.
