This paper presents a comprehensive study of statistical tests for comparing temporal point processes in general, with a particular focus on Poisson processes. We explore three key approaches: (1) an intensity-based test specifically for Poisson processes, (2) general parametric tests using the notion of maximum likelihood estimation, and (3) a general nonparametric test using the Isometric Log-Ratio (ILR) transformation. The first approach adopts a three-step procedure for comparing inhomogeneous Poisson processes by testing total and normalized intensities separately and then combining the corresponding p-values using Fisher’s method. The second method proposes a likelihood-based parametric test to examine the conditional intensity functions in point processes, emphasizing the application to Hawkes processes. Lastly, the third approach introduces a nonparametric test for general point processes, by transforming inter-event times into a Euclidean space via the ILR transformation, followed by conventional depth-based methods on multivariate data. We then conduct thorough studies on simulations as well as real-world data to illustrate these testing procedures and demonstrate their effectiveness.
Mu et al. (Thu,) studied this question.