I have read that a bootstrap distribution does not emulate the distribution under the null, and as such, we need to recenter our means if we want to do hypothesis testing using bootstrap (i.e. by subtracting out the sample means and then adding back in the mean under the null). However, do we need to do this (bootstrapping under the null) also if we are calculating confidence intervals?
I generally do not see this done on posted examples showing bootstrapped confidence intervals, but my knowledge is not yet deep enough to ascertain if this is valid. So, also, if we don't have to bootstrap under the null for confidence intervals, why not?