I am currently trying to get my head around some things concerning parametric bootstrap. Most things are probably trivial but I still think I may have missed something.
Suppose I want to get confidence intervals for data using a parametric bootstrap procedure.
So I have this sample and I assume its normally distributed. I would then estimate variance $\hat{v}$ and mean $\hat{m}$ and get my distribution estimate $\hat{P}$, which is obviously just $N(\hat{m},\hat{v})$.
Instead of sampling from that distribution I could just calculate the quantiles analytically and be done.
a) I conclude: in this trivial case, the parametric bootstrap would be the same as calculating things in a normal-distribution-assumption?
So theoretically this would be the case for all parametric bootstrap models, as long as I can handle the calculations.
b) I conclude: using the assumption of a certain distribution will bring me extra accuracy in the parametric bootstrap over the nonparametric one (if it is correct of course). But other than that, I just do it because I can't handle the analytic calculations und try to simulate my way out of it?
c) I would also use it if the calculations are "usually" done using some approximation because this would perhaps give me more accuracy...?
To me, the benefit of the (nonparametric) bootstrap seemed to lie in the fact that I don't need to assume any distribution. For the parametric bootstrap that advantage is gone - or are there things I've missed and where the parametric bootstrap provides a benefit over the things mentioned above?