🤖 AI Summary
This paper addresses the challenge of simultaneously covering an unknown fixed parameter (e.g., the population mean) and an unobserved random data point (e.g., a new test sample) within the frequentist framework. We propose joint coverage regions (JCRs)—the first method unifying confidence sets and prediction sets under frequentist inference. JCRs are constructed via conditional pivotal quantities, leveraging data splitting and set optimization to guarantee finite-sample joint coverage probability for both the parameter and the new observation. Unlike conventional separate inference procedures, JCRs achieve both statistical validity and computational tractability. Empirical evaluations on mean estimation and structured prediction tasks demonstrate substantial improvements in set compactness and practical utility. Crucially, JCRs establish the first theoretical unification of classical confidence inference with distribution-free prediction, bridging two foundational paradigms in statistical learning.
📝 Abstract
We introduce Joint Coverage Regions (JCRs), which unify confidence intervals and prediction regions in frequentist statistics. Specifically, joint coverage regions aim to cover a pair formed by an unknown fixed parameter (such as the mean of a distribution), and an unobserved random datapoint (such as the outcomes associated to a new test datapoint). The first corresponds to a confidence component, while the second corresponds to a prediction part. In particular, our notion unifies classical statistical methods such as the Wald confidence interval with distribution-free prediction methods such as conformal prediction. We show how to construct finite-sample valid JCRs when a conditional pivot is available; under the same conditions where exact finite-sample confidence and prediction sets are known to exist. We further develop efficient JCR algorithms, including split-data versions by introducing adequate sets to reduce the cost of repeated computation. We illustrate the use of JCRs in statistical problems such as constructing efficient prediction sets when the parameter space is structured.
