I was wondering how bootstrap CIs (and BCa in barticular) perform on normally-distributed data. There seems to be lots of work examining their performance on various types of distributions, but could not find anything on normally-distributed data. Since it seems an obvious thing to study first, I suppose the papers are just too old.
I did some Monte Carlo simulations using the R boot package and found bootstrap CIs to be in agreement with exact CIs, although for small samples (N<20) they tend to be a bit liberal (smaller CIs). For large enough samples, they are essentially the same.
This makes me wonder whether there is any good reason not to always use bootstrapping. Given the difficulty of assessing whether a distribution is normal, and the many pitfalls behind this, it seems reasonable not to decide and report bootstrap CIs irrespective of the distribution. I understand the motivation for not using non-parametric tests systematically, since they have less power, but my simulations tell me this is not the case for bootstrap CIs. They are even smaller.
A similar question that bugs me is why not always use the median as the measure of central tendency. People often recommend to use it to characterize non normally-distributed data, but since the median is the same as the mean for normally-distributed data, why make a distinction? It would seem quite beneficial if we could get rid of procedures for deciding whether a distribution is normal or not.
I am very curious about your thoughts on these issues, and whether they have been discussed before. References would be highly appreciated.