Monte Carlo approximation of bootstrap variances

It is widely believed that the number of resamples required for bootstrap variance estimation is relatively small. An argument based on the unconditional coefficient of variation of the Monte Carlo approximation, suggests that as few as 25 resamples will give reasonable results. In this article we argue that the number of resamples should, in fact, be determined by the conditional coefficient of variation, involving only resampling variability. Our conditional analysis is founded on a belief that Monte Carlo error should not be allowed to determine the conclusions of a statistical analysis and indicates that approximately 800 resamples are required for this purpose. The argument can be generalized to the multivariate setting and a simple formula is given For determining a lower bound on the number of resamples required to approximate an m-dimensional bootstrap variance-covariance matrix.