Why (mathematically) is the parametric bootstrap usually better than the empirical one? As I know from experience, the parametric bootstrap performs better in terms of coverage probability for confidence intervals then the empirical bootstrap.
Of course, this makes sense because you put in some information about the distribution and that is used to reduce the error.
However, thats not really a mathematical description of what is going on ...
Books such as Efron/Tibshirani don't go into details. Van der Vaarts Asymptotic Statistics only covers higher order consistency for studentized vs unstudentized intervals. I think Hall doesn't write anything about it either ...
Can I read up on this somewhere?
 A: Basically, all nonparametric bootstrap procedures underestimate the variance of the sampling distribution. This issue doesn't have a consistent name in the literature, but I like "narrowness bias" from "What Teachers Should Know About the Bootstrap: Resampling in the Undergraduate Statistics Curriculum" by Tim Hesterberg.
The reason the parametric bootstrap doesn't have this issue is because when you are building a parameterized model based on some data, you'll use the unbiased estimator for the variance (which is corrected by $ {n/(n-1)}$). Because the non-parametric bootstrap distribution is essentially the same thing as the plug-in estimate, its estimate of the variance of the sampling distribution (and thus the standard error) is too small. This is also why the issue goes away with very large sample sizes.
If you are stuck with small sample sizes, however, there are a two ways to get good non-parametric intervals. If you can ask your question in the form of a two-sample hypothesis test, inverting a permutation test will get you intervals with the proper coverage. If you can ask your question in the form of a one-sample hypothesis test, you will get decent intervals with a sign-change permutation test (or the wild bootstrap, which is essentially the same thing).
