基于极大似然估计的带标记Hawkes过程检验方法
Testing procedures based on maximum likelihood estimation for Marked Hawkes processes
AI总结：
针对带标记Hawkes过程复杂度提升带来的计算成本高、易过拟合问题，本文基于指数记忆核参数化Hawkes过程，拓展已有理论框架提出稳健的模型参数与复杂度检验方法，并通过数值研究验证其在理论不足场景下的实际稳健性。
摘要：
Hawkes模型是一种依赖历史的点过程，广泛应用于各领域的事件时间聚类建模。作为该框架的扩展，多维带标记Hawkes过程纳入了多种相互作用的事件类型与额外标记，提升了其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会增加相应估计方法的计算成本，还可能引发模型过拟合。因此，在模型的准确性与人为复杂度之间找到平衡至关重要。为找到合适的Hawkes过程版本，本文研究带标记Hawkes过程的模型拟合评估与参数检验任务。本文聚焦于带指数记忆核的参数化Hawkes过程——这是一种因理论与实践优势而被广泛使用的变体。本研究在已有理论框架的基础上进行拓展，提出了用于评估模型参数与复杂度的稳健检验方法。随后我们通过全面的数值研究验证了这些检验的实际稳健性，尤其是在理论保证尚不充分的场景下。
英文摘要：
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.
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