Say I run 100 entirely different estimation procedures on 100 different datasets and estimate 95% confidence intervals for each (e.g. logistic regression, linear regression, etc.). Purely based on the definition of a confidence interval, it seems like we'd expect ~95 of the intervals to capture the true parameter.
However, this is subtly different than the typical definition I've seen where confidence interval frequency properties are specific to an estimation procedure / parametric form.
Is there a good formal argument for and/or empirical example of why this cross-estimation procedure, cross-distribution frequency property holds?