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.