Hypothesis testing is a key part of empirical science and multiple testing as well as the combination of evidence from several tests are continued areas of research. In this article we consider the problem of combining the results of multiple hypothesis tests to i) test global hypotheses and ii) make marginal inference while controlling the k-FWER. We propose a new family of combination tests for joint hypotheses, which we show through simulation to have higher power than other combination tests against many alternatives. Furthermore, we prove that a large family of combination tests -- which includes the one we propose but also other combination tests -- admits a quadratic shortcut when used in a \CTP, which controls the FWER strongly. We develop an algorithm that is linear in the number of hypotheses for obtaining confidence sets for the number of false hypotheses among a collection of hypotheses and an algorithm that is cubic in the number of hypotheses for controlling the k-FWER for any k greater than one.

翻译：假设测试是实证科学和多种测试的关键部分,而若干测试的证据的结合是继续研究的领域。在本篇文章中,我们考虑了将多重假设测试的结果结合到(i)测试全球假设和(ii)在控制 k-FWER时进行边际推论的问题。我们建议对联合假设进行新的组合测试,我们通过模拟表明,对许多替代品的合并测试比其他组合测试具有更高的威力。此外,我们证明,一个大型组合测试组 -- -- 包括我们提议的测试,但也包括其他组合测试 -- -- 当用于一个严格控制FWER的\CTP时,会接受一种二次快捷的捷径。我们开发了一种算法,在假设数中是线性,以获得对一系列假假设中假假设数的信任,而算法则是用于控制K-FWER的任何大于1的假设数中的立项。