Allocation of Monte Carlo resources for the iterated bootstrap

Use of the iterated bootstrap is often recommended for calibration of bootstrap intervals, using either direct calibration of the nominal coverage probability (prepivoting), or additive correction of the interval endpoints. Monte Carlo resampling is a straightforward, ard, bur computationally expensive way to approximate the endpoints of bootstrap intervals. Booth and Hall examined the case of coverage calibration of Efron's percentile interval, and developed an asymptotic approximation for the error in the Monte Carlo approximation of the endpoints. Their results can be used to determine an approximately optimal allocation of resamples to the first and second level of the bootstrap. An extension of this result to the case of the additively corrected percentile interval shows that the bias of the Monte Carlo approximation to the additively corrected endpoints is of smaller order than in the case of direct coverage calibration, and the asymptotic variance is the same. Because the asymptotic bias is controlled by the number of second level resamples, and the asymptotic variance by the number of first level resamples, this indicates that comparable Monte Carlo accuracy can be achieved with far less computational effort for the additively corrected interval than for the coverage calibrated interval. For both methods of calibration, these results and supporting simulations show that, for an optimal allocation of computing resources, the number of second level resamples should generally be considerably less than the number of first level resamples. This is in contrast to the usual practice in the literature. Also, the number of first level resamples needed to achieve reasonable Monte Carlo accuracy for double bootstrap confidence intervals is roughly root 2 times greater than for single stage bootstrap confidence intervals, and again is generally underestimated in the literature.