Hawkes models are frequently used to describe point processes that are clustered spatial-temporally, and have been used in numerous applications including the study of earthquakes, invasive species, and contagious diseases. An extension of the Hawkes model is considered where the productivity is variable. In particular, the case is explored where each point may have its own productivity and a simple analytic formula is derived for the maximum likelihood estimators of these productivities. This estimator is compared with an empirical estimator and ways are explored of stabilizing both estimators by lower truncating, smoothing, and rescaling the estimates. Properties of the estimators are explored in simulations, and the methods are applied to seismological and epidemic datasets to show and quantify substantial variation in productivity.

中文翻译：

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可变生产力霍克斯过程的非参数估计

霍克斯模型经常用于描述时空聚集的点过程，并已用于许多应用，包括地震、入侵物种和传染病的研究。在生产率可变的情况下，考虑了霍克斯模型的扩展。特别是，探讨了每个点可能有自己的生产力的情况，并为这些生产力的最大似然估计推导出了一个简单的分析公式。将该估计量与经验估计量进行比较，并探索通过降低截断、平滑和重新调整估计量来稳定这两个估计量的方法。在模拟中探索了估计器的属性，并将这些方法应用于地震和流行病数据集，以显示和量化生产力的重大变化。