Joint Coverage Regions: Simultaneous Confidence and Prediction Sets

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.