The Hawkes model is a past-dependent point process, widely used in various fields for modeling temporal clustering of events. Extending this framework, the multidimensional marked Hawkes process incorporates multiple interacting event types and additional marks, enhancing its capability to model complex dependencies in multivariate time series data. However, increasing the complexity of the model also increases the computational cost of the associated estimation methods and may induce an overfitting of the model. Therefore, it is essential to find a trade-off between accuracy and artificial complexity of the model. In order to find the appropriate version of Hawkes processes, we address, in this paper, the tasks of model fit evaluation and parameter testing for marked Hawkes processes. This article focuses on parametric Hawkes processes with exponential memory kernels, a popular variant for its theoretical and practical advantages. Our work introduces robust testing methodologies for assessing model parameters and complexity, building upon and extending previous theoretical frameworks. We then validate the practical robustness of these tests through comprehensive numerical studies, especially in scenarios where theoretical guarantees remains incomplete.

翻译：霍克斯模型是一种依赖历史信息的点过程，广泛应用于各领域以建模事件的时间聚集性。在此框架基础上，多维标记霍克斯过程引入了多种相互关联的事件类型及附加标记，从而增强了对多元时间序列数据中复杂依赖关系的建模能力。然而，模型复杂度的提升也增加了相应估计方法的计算成本，并可能导致模型过拟合。因此，必须在模型精度与人为复杂度之间寻求平衡。为找到霍克斯过程的合适版本，本文致力于解决标记霍克斯过程的模型拟合评估与参数检验问题。本文重点研究具有指数记忆核的参数化霍克斯过程——该变体因其理论与实用优势而被广泛采用。我们在继承并拓展既有理论框架的基础上，提出了用于评估模型参数与复杂度的稳健检验方法。随后通过系统的数值研究验证了这些检验方法的实际鲁棒性，特别是在理论保证尚不完善的场景中。