The problem of quickest detection of a change in the mean of a sequence of independent observations is studied. The pre-change distribution is assumed to be stationary, while the post-change distributions are allowed to be non-stationary. The case where the pre-change distribution is known is studied first, and then the extension where only the mean and variance of the pre-change distribution are known. No knowledge of the post-change distributions is assumed other than that their means are above some pre-specified threshold larger than the pre-change mean. For the case where the pre-change distribution is known, a test is derived that asymptotically minimizes the worst-case detection delay over all possible post-change distributions, as the false alarm rate goes to zero. Towards deriving this asymptotically optimal test, some new results are provided for the general problem of asymptotic minimax robust quickest change detection in non-stationary settings. Then, the limiting form of the optimal test is studied as the gap between the pre- and post-change means goes to zero, called the Mean-Change Test (MCT). It is shown that the MCT can be designed with only knowledge of the mean and variance of the pre-change distribution. The performance of the MCT is also characterized when the mean gap is moderate, under the additional assumption that the distributions of the observations have bounded support. The analysis is validated through numerical results for detecting a change in the mean of a beta distribution. The use of the MCT in monitoring pandemics is also demonstrated.
翻译:研究的是最快速地发现一个独立观测序列平均值变化的问题; 假设变化前分布是固定的, 而变化后分布则允许是非静止的; 首先研究知道变化前分布的情况, 然后研究仅知道变化前分布的平均值和差异的扩展情况; 假设对变化后分布情况了解最迅速的问题, 只是假设其手段超过某些预先规定的临界值, 大于变化前分布值; 对于已知变化前分布的情况, 测试的结果是, 尽可能地将所有可能的变化后分布的最坏情况检测延迟减少到最低程度, 因为错误的警报率达到零; 之后研究的是, 对变化前分布情况进行最坏的检查; 得出一些新的结果, 由此得出关于变化前分布的无症状微缩缩缩微缩和最强的快速变化检测问题; 然后, 研究中度测试的有限形式, 是因为变化前和变化后手段之间的差距达到零, 也称对变化后分布情况进行最坏的检测结果进行最坏的检测; 在进行误测测测时, 也显示M- 变的数值分析中, 也显示, 数值的数值变变变的数值是 。