A signal recovery problem is considered, where the same binary testing problem is posed over multiple, independent data streams. The goal is to identify all signals, i.e., streams where the alternative hypothesis is correct, and noises, i.e., streams where the null hypothesis is correct, subject to prescribed bounds on the classical or generalized familywise error probabilities. It is not required that the exact number of signals be a priori known, only upper bounds on the number of signals and noises are assumed instead. A decentralized formulation is adopted, according to which the sample size and the decision for each testing problem must be based only on observations from the corresponding data stream. A novel multistage testing procedure is proposed for this problem and is shown to enjoy a high-dimensional asymptotic optimality property. Specifically, it achieves the optimal, average over all streams, expected sample size, uniformly in the true number of signals, as the maximum possible numbers of signals and noises go to infinity at arbitrary rates, in the class of all sequential tests with the same global error control. In contrast, existing multistage tests in the literature are shown to achieve this high-dimensional asymptotic optimality property only under additional sparsity or symmetry conditions. These results are based on an asymptotic analysis for the fundamental binary testing problem as the two error probabilities go to zero. For this problem, unlike existing multistage tests in the literature, the proposed test achieves the optimal expected sample size under both hypotheses, in the class of all sequential tests with the same error control, as the two error probabilities go to zero at arbitrary rates. These results are further supported by simulation studies and extended to problems with non-iid data and composite hypotheses.
翻译:考虑信号恢复问题, 因为在多个独立的数据流中也存在相同的二进制测试问题。 目标是确定所有信号, 即替代假设准确的流流, 和噪音, 即空假设正确, 遵循传统或通用家族错误概率的指定界限。 不需要事先知道信号的确切数量, 仅假设信号和噪音数量的上限。 采用了分散化的配方, 根据这种配方, 每个测试问题的样本大小和决定必须仅基于对应数据流的观测。 为这一问题提议了一个全新的多阶段测试程序, 并且显示无空假设的流符合传统或通用家庭错误概率的指定界限。 具体地说, 它实现了所有流的平均平均值, 预期的样本大小, 与信号的真实数量一致, 因为信号和噪声的最大数量以任意率进入不精确度。 在所有顺序测试中, 都采用相同的全球错误控制的等级, 。 相比之下, 文献中现有的多阶段测试过程测试程序, 其现有的不透明性测试, 只能通过直径直的两部测试结果, 以直径直为直径, 基础测试结果的预测结果 。 这些测试中, 以直径直径直径直径为直为直为直为直度测试速度, 。 。 这些测试结果研究 以直为直为直为直为直为直为直为直为直为直度测试结果, 。 。 。 。 。 。 。 。 根根根根根根根根根根根根根根根根根根根根根根根根根根底, 。