We study a large-scale one-sided multiple testing problem in which test statistics follow normal distributions with unit variance, and the goal is to identify signals with positive mean effects. A common approach is to compute $p$-values under the assumption that all null means are exactly zero and then apply standard multiple testing procedures such as the Benjamini--Hochberg (BH) or Storey--BH method. However, because the null hypothesis is composite, some null means may be strictly negative. In this case, the resulting $p$-values are conservative, leading to a substantial loss of power. Existing methods address this issue by modifying the multiple testing procedure itself, for example through conditioning strategies or discarding rules. In contrast, we focus on correcting the $p$-values so that they are exact under the null. Specifically, we estimate the marginal null distribution of the test statistics within an empirical Bayes framework and construct refined $p$-values based on this estimated distribution. These refined $p$-values can then be directly used in standard multiple testing procedures without modification. Extensive simulation studies show that the proposed method substantially improves power when $p$-values are conservative, while achieving comparable performance to existing methods when $p$-values are exact. An application to phosphorylation data further demonstrates the practical effectiveness of our approach.

翻译：我们研究一个大规模单侧多重检验问题，其中检验统计量服从单位方差的正态分布，目标是识别具有正均值效应的信号。一种常见方法是在所有零假设均值恰好为零的假设下计算$p$值，然后应用标准的多重检验程序，如Benjamini--Hochberg（BH）或Storey--BH方法。然而，由于零假设是复合的，某些零假设均值可能严格为负。在这种情况下，所得的$p$值是保守的，导致检验功效显著下降。现有方法通过修改多重检验程序本身来解决此问题，例如通过条件策略或舍弃规则。相比之下，我们专注于修正$p$值，使其在零假设下是精确的。具体而言，我们在经验贝叶斯框架内估计检验统计量的边际零分布，并基于此估计分布构建改进的$p$值。这些改进的$p$值无需修改即可直接用于标准多重检验程序。大量的模拟研究表明，当$p$值保守时，所提方法能显著提升检验功效，而在$p$值精确时，其性能与现有方法相当。在磷酸化数据上的应用进一步证明了我们方法的实际有效性。