confidence intervals about confidence intervals

[gusl]

Over lunch on Friday, Michael Littman told me the following idea: Suppose you have a procedure for making confidence intervals out of an IID sample from a population. If you run this sample/estimate cycle multiple times, the resulting confidence intervals should mostly overlap with each other. This is a coherence constraint, which I'll try to make more precise (later), the intuition being: the higher the confidence levels, the more they should overlap (by some measure).

This also suggests a bootstrap procedure to produce multiple confidence intervals, as a way to approximate the sampling distribution of the "confidence interval" statistic.

Student (a.k.a. William Sealy Gosset) has supposed dealt with these questions, including some question about infinite regress, but I haven't yet found the reference.