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:我们引入了联合覆盖区(JCRs),在常年统计数字中统一了信任间隔和预测区域,具体地说,联合覆盖区旨在覆盖由未知固定参数(如分布平均值)和未观测随机数据点(如与新测试数据点相关的结果)组成的对子(如与新测试数据点有关的结果)构成的对子(JCRs),第一个对应了信任部分,第二个对应了预测部分。特别是,我们的概念统一了典型的统计方法,如Wald信任间隔,与无分布式预测方法(如符合预测)的对子。我们展示了如何在有条件的轴心时构建有限抽样有效联合计算;在同样的条件下,已知存在精确的有限抽样信心和预测数据集。我们进一步开发了高效的 JCR 算法,包括通过引入适当组合来降低重复计算的成本,包括多发数据版本。我们举例说明了在统计问题中使用的 JCRs,例如当参数空间结构时构建高效的预测组。</s>