Motivated by applications in cybersecurity and epidemiology, we consider the problem of detecting an abrupt change in the intensity of a Poisson process, characterised by a jump (non transitory change) or a bump (transitory change) from constant. We propose a complete study from the nonasymptotic minimax testing point of view, when the constant baseline intensity is known or unknown. The question of minimax adaptation with respect to each parameter (height, location, length) of the change is tackled, leading to a comprehensive overview of the various minimax separation rate regimes. We exhibit three such regimes and identify the factors of the two phase transitions, by giving the cost of adaptation to each parameter. For each alternative hypothesis, depending on the knowledge or not of each change parameter, we propose minimax or minimax adaptive tests based on linear statistics, close to CUSUM statistics, or quadratic statistics more adapted to the L 2-distance considered in our minimax criteria and typically more powerful in practice, as our simulation study shows. When the change location or length is unknown, our adaptive tests are constructed from a scan aggregation principle combined with Bonferroni or min-p level correction, and a conditioning trick when the baseline intensity is unknown.
翻译:以网络安全和流行病学的应用为动力,我们考虑发现Poisson进程强度突变的问题,其特点是跳跃(非过渡性变化)或从恒定中跳动(短暂变化)或跳跃(短暂变化),发现Poisson进程强度突变的问题。我们建议从非同步微型最大试验点的角度进行一项完整的研究,当恒定基线强度为已知或未知时,我们建议从非同步小型最大试验点进行一项完整的研究。解决了对变化的每个参数(高度、位置、长度)进行微缩最大适应的问题,导致对各种微缩最大分离率制度进行全面的概览。我们展示了三种这样的制度,并通过给每个参数提供适应成本来确定两个阶段过渡的因素。对于每一种替代假设,我们建议根据线性统计、接近CUSUM统计、或更适应性微缩缩图统计数据,或更适应我们微缩图中考虑的L 2-距离标准以及通常更强大的实践,正如我们的模拟研究表明的那样,在变化地点或长度不明时,我们的适应性测试是从扫描组合原则中构建一个未知的变压底基度和微调度的底压级校正。