My question is firstly wondering about the correct procedure on how to bootstrap under the null, and secondly more or less re-asking the following question which has not received any answers: Why shift the mean of a bootstrap distribution when conducting a hypothesis test?
If we use the bootstrap for hypothesis testing, some applied researchers impose the Null Hypothesis on the empirical distribution (which typically involves some manipulation of the empirical distribution). While I do understand why this might sound appealing (enforcing consistency with your theory), I'm having some problems understanding the steps and the consequences of this.
Suppose we have the simple case of inference about the sample mean. I have $H_0: E[X]=0$. If I resample under the null, I would then center my distribution on 0. But in that case, there seems to be no way to reject $H_0$, since we sample from a distribution that is enforced to be mean 0?
Is there any theoretical justification other than internal consistency for resampling under the null? Does it have any adverse consequences (other than being more complicated than not doing this)?
