论文标题
异常假设检验的渐近学
Asymptotics for Outlier Hypothesis Testing
论文作者
论文摘要
我们重新审视了li \ emph {et al。}（tit 2014）的异常假设检验框架，并得出了最佳测试的基本限制。在异常假设检验中，给出了一个观察到的序列，其中大多数序列都是生成i.i.d的。来自名义分布。任务是辨别根据异常分布生成的外围序列集。名义和异常分布是\ emph {unknown}。我们考虑了多个异常值的情况，在这种情况下，异常值的数量未知，并且每个异常值都可以遵循不同的异常分布。在这种情况下，我们研究了错误分类错误，错误警报和错误拒绝的概率之间的权衡。具体来说，我们提出了一个基于阈值的测试，以确保错误分类错误和错误警报概率的指数衰减。我们研究了对错误拒绝概率的两个限制，其中一个约束是它是一个不变常数，另一个是它具有指数衰减率。在这两种情况下，我们都会针对每个名义和异常分布的元组来表征错误拒绝概率的界限，这是阈值的函数。最后，我们在广义的Neyman-Pearson标准下证明了测试的渐近最佳性。
We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated according to anomalous distributions. The nominal and anomalous distributions are \emph{unknown}. We consider the case of multiple outliers where the number of outliers is unknown and each outlier can follow a different anomalous distribution. Under this setting, we study the tradeoff among the probabilities of misclassification error, false alarm and false reject. Specifically, we propose a threshold-based test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probability, with one constraint being that it is a non-vanishing constant and the other being that it has an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each tuple of nominal and anomalous distributions. Finally, we demonstrate the asymptotic optimality of our test under the generalized Neyman-Pearson criterion.
