I confess to being guilty of this. My question: would we even be able to say an N% bootstrap “confidence” interval has this intuitive interpretation? It is based on actual repeated sampling of the observed data, and via the ‘plug-in principle’ that justifies the bootstrap, I cannot see why the bootstrap interpretation would be an error.

Could someone clarify this distinction (if any) between the classically computed interval, and the bootstrap one?

Update: Relevant paper by Efron and DiCiccio on Bootstrap intervals. He alludes to problems with intervals derived from asymptotic theory, that can be addressed using the bootstrap.

https://projecteuclid.org/euclid.ss/1032280214#abstract

Equation 1.1 refers to:

\hat{\theta} \pm z^{(\alpha)}\hat{\sigma}

Blockquote

The trouble with standard intervals is that they are based on an asymptotic approximation that can be quite inaccurate in practice. The example below illustrates what every applied statistician knows, that (1.1) can considerably differ from exact intervals in those cases where exact intervals exist. Over the years statisticians have developed tricks for improving (1.1), involving bias-corrections and parameter transformations. The bootstrap conﬁdence intervals that we will discuss here can be thought of as automatic algorithms for carrying out these improvements without human intervention.

Update 2: Empirical evaluation of classic vs bootstrap intervals from statisticians at U.S. Forest Service

Scanning the data, the differences in coverage were really small, but classic intervals were generally better.

Related to the prediction interval in this thread:

I’ve been driving myself mad trying to figure out a mathematical justification for this result. The best I can come up with so far is:

Upper bound on 2 random 95% intervals is

0.95^2 = 0.9025

Lower bound for particular interval from planned experiment = CI*Power, where:

CI = 0.95, Power = 0.8

Probability of future interval covering parameter

0.95 * 0.8 = 0.76

Averaging UB and LB:

\frac{1}{2}(0.9025 + 0.76) = 0.83125

I tried looking up the paper, but do not have access to it at the moment. Has someone read it? Did they give a mathematical justification there? Thanks.