Printed in Great Britain Fast and accurate approximate double bootstrap confidence intervals

We propose two new methods for the construction of approximate iterated bootstrap confidence intervals. Our methods follow a saddlepoint approach, but employ approxi-mate rather than exact solutions to the nonlinear saddlepoint equations that arise in applying that technique. The main advantages of the new techniques are that they yield confidence intervals with high coverage accuracy in small- to moderately-sized samples and that they achieve this accuracy with very substantial reductions in computation time compared to other methods. We illustrate our methods on calculations of confidence intervals for a ratio of means and for a correlation coefficient