Testing procedures based on maximum likelihood estimation for Marked Hawkes processes

TL;DR

This paper develops and validates robust testing procedures for evaluating the fit and complexity of parametric marked Hawkes processes, balancing model accuracy with computational efficiency in multivariate event data modeling.

Contribution

It introduces new statistical tests for model parameters and complexity in marked Hawkes processes, extending existing theoretical frameworks and validating them through numerical experiments.

Findings

New testing methodologies for Hawkes process parameters

Validated robustness of tests through numerical simulations

Provided practical tools for model selection in complex event data

Abstract

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.