Simultaneous statistical inference has been a cornerstone in the statistics methodology literature because of its fundamental theory and paramount applications. The mainstream multiple testing literature has traditionally considered two frameworks: the sample size is deterministic, and the test statistics corresponding to different tests are independent. However, in many modern scientific avenues, these assumptions are often violated. There is little study that explores the multiple testing problem in a sequential framework where the test statistics corresponding to the various streams are dependent. This work fills this gap in a unified way by considering the classical means-testing problem in an equicorrelated Gaussian and sequential framework. We focus on sequential test procedures that control the type I and type II familywise error probabilities at pre-specified levels. We establish that our proposed test procedures achieve the optimal expected sample sizes under every possible signal configuration asymptotically, as the two error probabilities vanish at arbitrary rates. Towards this, we elucidate that the ratio of the expected sample size of our proposed rule and that of the classical SPRT goes to one asymptotically, thus illustrating their connection. Generalizing this, we show that our proposed procedures, with appropriately adjusted critical values, are asymptotically optimal for controlling any multiple testing error metric lying between multiples of FWER in a certain sense. This class of metrics includes FDR/FNR, pFDR/pFNR, the per-comparison and per-family error rates, and the false positive rate.

翻译：同时统计推断因其基础性理论和重要应用而成为统计方法论文献的基石。传统多重检验文献主要考虑两个框架：样本量确定，且不同检验对应的检验统计量相互独立。然而，在现代科学研究的诸多前沿领域中，这些假设常常被违反。目前鲜有研究探讨在序贯框架下，各数据流对应检验统计量存在相依性时的多重检验问题。本文通过考虑等相关高斯序贯框架下的经典均值检验问题，以统一方式填补了这一空白。我们聚焦于能够将族系第一类与第二类错误概率控制在预设水平的序贯检验程序。研究表明，当两类错误概率以任意速率趋近于零时，所提出的检验程序能在所有可能的信号配置下渐近达到最优期望样本量。为此，我们阐明所提规则与经典SPRT的期望样本量之比渐近趋于1，从而揭示二者间联系。推广而言，我们证明了所提出的程序在适当调整临界值后，对控制某种意义下介于FWER倍数之间的任意多重检验误差指标具有渐近最优性。这类指标包括FDR/FNR、pFDR/pFNR、每比较错误率、每族错误率以及假阳性率。