We propose a new testing framework applicable to both the two-sample problem on point processes and the community detection problem on rectangular arrays of point processes, which we refer to as longitudinal networks; the latter problem is useful in situations where we observe interactions among a group of individuals over time. Our framework is based on a multiscale discretization scheme that consider not just the global null but also a collection of nulls local to small regions in the domain; in the two-sample problem, the local rejections tell us where the intensity functions differ and in the longitudinal network problem, the local rejections tell us when the community structure is most salient. We provide theoretical analysis for the two-sample problem and show that our method has minimax optimal power under a Holder continuity condition. We provide extensive simulation and real data analysis demonstrating the practicality of our proposed method.

翻译：本文提出了一种新的检验框架，适用于点过程的两样本问题以及点过程矩形阵列（我们称之为纵向网络）的社区检测问题；后一问题在观测群体个体随时间交互作用时具有重要应用价值。我们的框架基于多尺度离散化方案，不仅考虑全局零假设，还考虑定义域内小区域对应的局部零假设集合：在两样本问题中，局部拒绝区域指示强度函数的差异位置；在纵向网络问题中，局部拒绝区域揭示社区结构最显著的时间段。我们对两样本问题进行了理论分析，证明在Hölder连续性条件下该方法具有极小极大最优功效。通过大量仿真与真实数据分析，验证了所提方法的实用性。