An Example of Slow Convergence of the Bootstrap in High Dimensions

This article examines the use of bootstrap hypothesis tests for testing the equality of two multivariate distributions. The test statistic used is the maximum of the univariate two-sample t-statistics. Depending upon the type of bootstrap resampling used, the simulation studies show that the test levels are conservative or anti-conservative when the sample sizes are small and the number of variables is large. For small sample sizes, using the bootstrap resampling that preserves the Type I error can lead to a testing procedure that has lower power, sometimes dramatically lower, than a permutation test.