Space–time inhomogeneous background intensity estimators for semi-parametric space–time self-exciting point process models


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.

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The authors would like to express their sincere gratitude to the reviewers and editor for very helpful suggestions and comments which greatly improved this paper. The authors would like to thank the National Natural Science Foundation of China (No. 91746107, 91746205); the State Scholarship Fund of China Scholarship Council (CSC); and the National Science and Engineering Research Council (NSERC) of Canada, for their funding and support.

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Correspondence to Zhanjie Song.

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Li, C., Song, Z. & Wang, W. Space–time inhomogeneous background intensity estimators for semi-parametric space–time self-exciting point process models. Ann Inst Stat Math 72, 945–967 (2020).

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  • Space–time point process models
  • Kernel density estimation
  • Expectation–maximization algorithm
  • Maximum likelihood