Histogram maximum likelihood estimators of semi-parametric space–time self-exciting point process models via expectation–maximization algorithm can be biased when the background process is inhomogeneous. We explore an alternative estimation method based on the variable bandwidth kernel density estimation (KDE) and EM algorithm. The proposed estimation method involves expanding the semi-parametric models by incorporating an inhomogeneous background process in space and time and applying the variable bandwidth KDE to estimate the background intensity function. Using an example, we show how the variable bandwidth KDE can be estimated this way. Two simulation examples based on residual analysis are designed to evaluate and validate the ability of our methods to recover the background intensity function and parametric triggering intensity function.

中文翻译：

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半参数时空自激点过程模型的时空非均匀背景强度估计

当背景过程不均匀时，通过期望最大化算法的半参数时空自激点过程模型的直方图最大似然估计量可能存在偏差。我们探索了一种基于可变带宽核密度估计 (KDE) 和 EM 算法的替代估计方法。所提出的估计方法涉及通过在空间和时间上结合非均匀背景过程并应用可变带宽 KDE 来估计背景强度函数来扩展半参数模型。通过一个例子，我们展示了如何以这种方式估计可变带宽 KDE。设计了两个基于残差分析的模拟示例来评估和验证我们的方法恢复背景强度函数和参数触发强度函数的能力。