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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?

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