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 co verage probability (prepivoting), or additive correction of the interv al endpoints. Monte Carlo resampling is a straightforward, ard, bur co mputationally 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 approximatio n for the error in the Monte Carlo approximation of the endpoints. The ir results can be used to determine an approximately optimal allocatio n of resamples to the first and second level of the bootstrap. An exte nsion of this result to the case of the additively corrected percentil e interval shows that the bias of the Monte Carlo approximation to the additively corrected endpoints is of smaller order than in the case o f direct coverage calibration, and the asymptotic variance is the same . Because the asymptotic bias is controlled by the number of second le vel resamples, and the asymptotic variance by the number of first leve l resamples, this indicates that comparable Monte Carlo accuracy can b e achieved with far less computational effort for the additively corre cted interval than for the coverage calibrated interval. For both meth ods of calibration, these results and supporting simulations show that , for an optimal allocation of computing resources, the number of seco nd level resamples should generally be considerably less than the numb er of first level resamples. This is in contrast to the usual practice in the literature. Also, the number of first level resamples needed t o achieve reasonable Monte Carlo accuracy for double bootstrap confide nce intervals is roughly root 2 times greater than for single stage bo otstrap confidence intervals, and again is generally underestimated in the literature.