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.

翻译：我们提出联合覆盖区域（JCRs），它统一了频率统计中的置信区间与预测区域。具体而言，联合覆盖区域旨在覆盖由未知固定参数（如分布均值）与未观测随机数据点（如新测试数据点对应的结果）构成的配对。前者对应置信分量，后者对应预测部分。特别地，我们的概念统一了经典统计方法（如Wald置信区间）与无分布预测方法（如共形预测）。我们展示了在条件枢轴量存在时——即已知精确有限样本置信集与预测集存在的相同条件下——如何构造有限样本有效的JCRs。我们进一步开发了高效的JCR算法，包括通过引入适当集合降低重复计算代价的分割数据版本。我们通过参数空间结构化时构造高效预测集等统计问题，演示了JCRs的应用。