# How to calculate confidence intervals using subsampling after a nonparametric estimator about the empirical distribution function?

I have a problem where I think subsampling is more appropriate than the bootstrap. (Reason in another post.)

However, I found no quick reference on subsampling CIs, and my naive inversion of the basic theory led to something that looks like the analogue of the "basic bootstrap" CI which seems to be used less often itself. Also, another question got a reply that subsampling applications usually "rely on asymptotic normality of the estimate", and "a bootstrap of some sort is more appropriate eg for asymmetric confidence intervals."

My (limited) understanding of subsampling does not seem to imply that.

I think I can get CIs as $$(\theta -\frac{\sqrt{b}}{\sqrt{n}}(\theta^{*}_{(1-\alpha)}-\theta);\theta -\frac{\sqrt{b}}{\sqrt{n}}(\theta^{*}_{(\alpha)}-\theta))$$ where $\theta^{*}_{(1-\alpha)}$ denotes the $1-\alpha$ percentile of the subsampled coefficients $\theta^{*}$ and $\theta$ is original estimate (so yes, everything could use $\hat{\theta}$ here).

So, if I have a nonparametric estimator for the "excess mass" at some parts of the distribution (implicitly using kernel density estimates usually unfit for the bootstrap), and I "simply" repeat the process with subsamples without replacement, how do I get a CI from the distribution of estimates? Take the raw percentiles (á la "percentile bootstrap") or invert the statistic from the general theory ("basic bootstrap").

Monte Carlos would be prohibitively expensive to compare these.

By the way, I know subsampling is not really a smart move without deriving something about the rate of convergence (simple square root above) and the right method to pick the size of the subsample (b). But is a "percentile bootstrap" any smarter in this case?

Finally, maybe a question worthy of its own posting, but if "b=n" is not required for the common bootstrap either (nor is no-replacement, thus subsampling could be a subset of bootstrap methods), why isn't the $\frac{\sqrt{b}}{\sqrt{n}}$ correction part of the bootstrap CI formulas (basic or BCa)?