The effect of Monte Carlo approximation on coverage error of double-bootstrap confidence intervals

A double-bootstrap confidence interval must usually be approximated by a Mo nte Carlo simulation, consisting of two nested levels of bootstrap sampling . We provide an analysis of the coverage accuracy of the interval which tak es account of both the inherent bootstrap and Monte Carlo errors. The analy sis shows that, by a suitable choice of the number of resamples drawn at th e inner level of bootstrap sampling, we can reduce the order of coverage er ror. We consider also the effects of performing a finite Monte Carte simula tion on the mean length and variability of length of two-sided intervals. A n adaptive procedure is presented for the choice of the number of inner lev el resamples. The effectiveness of the procedure is illustrated through a s mall simulation study.