The problem of quickest detection of a change in the distribution of a sequence of independent observations is considered. It is assumed that the pre-change distribution is known (accurately estimated), while the only information about the post-change distribution is through a (small) set of labeled data. This post-change data is used in a data-driven minimax robust framework, where an uncertainty set for the post-change distribution is constructed using the Wasserstein distance from the empirical distribution of the data. The robust change detection problem is studied in an asymptotic setting where the mean time to false alarm goes to infinity, for which the least favorable post-change distribution within the uncertainty set is the one that minimizes the Kullback-Leibler divergence between the post- and the pre-change distributions. It is shown that the density corresponding to the least favorable distribution is an exponentially tilted version of the pre-change density and can be calculated efficiently. A Cumulative Sum (CuSum) test based on the least favorable distribution, which is referred to as the distributionally robust (DR) CuSum test, is then shown to be asymptotically robust. The results are extended to the case where the post-change uncertainty set is a finite union of multiple Wasserstein uncertainty sets, corresponding to multiple post-change scenarios, each with its own labeled data. The proposed method is validated using synthetic and real data examples.

翻译：本文研究独立观测序列分布发生变化的快速检测问题。假设变化前分布已知（可精确估计），而关于变化后分布的唯一信息来源是一组（小规模）标记数据。在数据驱动的极小极大鲁棒框架中，利用该变化后数据构建基于Wasserstein距离的经验分布不确定集，以刻画变化后分布的不确定性。在平均虚警时间趋于无穷的渐近场景下研究鲁棒变化检测问题，此时不确定集中最不利的变化后分布应是使变化前后分布之间的Kullback-Leibler散度最小化的分布。研究表明，该最不利分布对应的概率密度是变化前密度的指数倾斜形式，且可高效计算。基于最不利分布的累积和（CuSum）检验（称为分布鲁棒CuSum检验）被证明具有渐近鲁棒性。进一步将结果推广至变化后不确定集为多个Wasserstein不确定集有限并集的情形，每个子集对应具有各自标记数据的多种变化后场景。通过合成数据与真实数据实例验证了所提方法的有效性。