In this paper, we consider the problem of detecting signals in multiple, sequentially observed data streams, where the distribution of each stream lies in one of two common composite spaces, depending on whether it is a signal or a noise. For this problem, we study a practical yet underexplored setting where it is a priori known that all signals have an identical distribution and so do all noises. Compared to the general setting where local distributions are free to take any values, this structure facilitates faster decision-making thanks to a smaller joint distribution space. However, it introduces additional challenges to the analysis of problem and design of tests, since the local distributions are now coupled. In this paper, we first establish a universal lower bound on the minimum expected sample size, which characterizes the essential difficulty of the problem and involves constants that are neither the minimum Kullback-Leibler divergences between the signal/noise distribution to the noise/signal distribution space, which appear in the lower bound for the general setting, nor the Kullback-Leibler divergences between the signal distribution and the noise distribution. Besides, we propose a test that controls the two types of familywise error rates below arbitrary levels, and achieves the minimum expected sample size asymptotically as the levels go to zero. Numerical studies are presented to compare with the state-of-the-art test for the general setting and demonstrate robustness against model misspecification.

翻译：本文考虑在多个顺序观测数据流中检测信号的问题，其中每个数据流的分布取决于其属于信号还是噪声，分别取自两个共同的复合分布空间之一。针对该问题，我们研究了一个实用但尚未充分探索的场景：先验已知所有信号具有相同分布，所有噪声亦具有相同分布。与局部分布可自由取任意值的一般场景相比，这种结构因联合分布空间更小而有利于加速决策。然而，由于局部分布间存在耦合关系，这给问题分析与检验设计带来了额外挑战。本文首先建立了最小期望样本量的通用下界，该下界刻画了问题的本质难度，其包含的常数既非一般场景下界中出现的信号/噪声分布与噪声/信号分布空间之间的最小KL散度，也非信号分布与噪声分布之间的KL散度。此外，我们提出了一种检验方法，能够将两类族系错误率控制在任意指定水平以下，并在错误率趋于零时渐近达到最小期望样本量。最后通过数值研究展示其与一般场景下最新检验方法的对比，并验证了对模型误设的鲁棒性。