# Confidence interval determination for skewed bootstrap parameter distribution

I have been reading around the literature and have been trying to work out the correct way (or most accurate way) to calculate a 68.3% confidence interval using bootstrapping for my particular data sample, but wasn't 100% clear so far.

I have a bootstrapped parameter distribution that is non-normal, and has a definite skew to the right (see attached image). It has been suggested to me to simply determine the confidence interval (which will be asymmetric around the mean in this case) by removing N * 0.5 * (1-0.683) of the N bootstrapping results from left and right and then taking these end points as the 15.85% and 84.15% quantiles. From reading around, it seems that for skewed and/or biased bootstrap parameter distributions (as is the case here) I should instead use the BCa bootstrap interval to determine the confidence intervals, as this will provide more accurate intervals with better coverage for this particular situation than the nominal method described above (which I think assumes normality of the parameter distribution?)

Is this the correct interpretation, and if so could someone please explain to me why?

If you can assume normality, which means that the distribution is determined by $$\mu$$ and $$\sigma$$, it is better to estimate these values (e.g. by means of the bootstrap) and base the confidence intervals on these estimators, i.e. $$\mu\pm z_{1-\alpha/2}\sigma$$).
If you cannot assume normality, your approach of simply taking the percentiles of the bootstraped computations of the observable is perfectly valid and known as the (non-parametric) percentile bootstrap interval. This method does not assume normality of the parameter distribution, but for unsymmetric distributions, there are examples where this confidence interval does not have a good coverage probability for smaller $$n$$. The $$BC_a$$ bootsrap interval ("bias corrected accelerated") compensates for this and typically has better coverage probability in such cases.
Out of curiosity, I had made Monte Carlo simulations to compare the different (non-parametric) bootstrap intervals, and the $$BC_a$$ bootstrap interval indeed had much faster convergence to the nominal coverage probability (see figures 7(a) and 8(a)):