A Note on Bootstrap Variance Estimation

The bootstrap estimator of the asymptotic covariance matrix of a function of sample means or sample quantiles is inconsistent in some situations. A modified bootstrap estimator is proposed and shown to be consistent under weak conditions. A simulation study shows that in terms of finite-sample performance, the improvement of this modification is substantial. The computation of this modified bootstrap estimator is much easier and cheaper than that of the estimator based on the quantiles of the bootstrap distribution. It is shown by simulation that with the same number of bootstrap replicates in bootstrap Monte Carlo approximation, the modified bootstrap estimator is more accurate than the estimator based on the interquartile range of the bootstrap distribution.