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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?

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    $\begingroup$ Is it really better? If you simulated a normal sample and did a parametric bootstrap assuming normal distribution, then of course you are in a perfect world. But if you simulated an exponentially distributed sample and did a parametric bootstrap assuming gamma or Weibull, would the parametrics bootstrap still be better? What if you assumed lognormal or, worse yet, normal? I personally think parametric bootstrap is your last resort when you cannot get to the empirical distribution of interest (e.g., in multilevel and small area models, DOI: 10.1111/j.1467-9868.2006.00541.x). $\endgroup$
    – StasK
    Aug 14, 2014 at 16:54
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    $\begingroup$ Well i guess, of course, it depends - in that perfect world, it does. For my BSc thesis i simulated from a bivariate normal distribution and calculated different CIs for the correlation coefficient to compare actual coverage. With Fisher Transformation, i get the nominal 95%, with parametric bootstrap, i get around 93% and with nonparametric its about 91%, BCa is 92%. But using the (real world) lawschool dataset from the bootstrap package, the differences get a bit smaller but parametric is still the best bootstrap. $\endgroup$
    – BootstrapBill
    Aug 16, 2014 at 8:39
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    $\begingroup$ Can you explain how you determined the best bootstrap for the lawschool dataset? $\endgroup$
    – eric_kernfeld
    Feb 17, 2018 at 23:31
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    $\begingroup$ Can you be a bit more specific? It's been a while but I could have look at my thesis (unfortunately it's in german, otherwise I'd offer to send it) $\endgroup$
    – BootstrapBill
    Feb 20, 2018 at 8:56

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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).

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