On the Number of Bootstrap Simulations Required to Construct a Confidence Interval

We make two points about the number, $B$ of bootstrap simulations needed to construct a percentile-$t$ confidence interval based on an $n$ sample from a continuous distribution: (i) The bootstrap's reduction of error of coverage probability, from $O(n^{-1/2})$ to $O(n^{-1})$, is available uniformly in $B$, provided nominal coverage probability is a multiple of $(B + 1)^{-1}$. In fact, this improvement is available even if the number of simulations is held fixed as $n$ increases. However, smaller values of $B$ can result in longer confidence intervals. (ii) In a large sample, the simulated statistic values behave like random observations from a continuous distribution, unless $B$ increases faster than any power of sample size. Only if $B$ increases exponentially quickly with $n$ is there a detectable effect due to discreteness of the bootstrap statistic.