Abstract
This paper presents a parametric estimation method for ill-observed linear stationary Hawkes processes. When the exact locations of points are not observed, but only counts over time intervals of fixed size, methods based on the likelihood are not feasible. We show that spectral estimation based on Whittle’s method is adapted to this case and provides consistent and asymptotically normal estimators, provided a mild moment condition on the reproduction function. Simulated data sets and a case-study illustrate the performances of the estimation, notably of the reproduction function even when time intervals are relatively large.
Acknowledgments
The authors would like to thank François Roueff who suggested the use of Whittle’s method for the estimation of Hawkes processes from bin-count data and Theorem 1’s extension to exponentially decaying reproduction kernels. The authors would also like to thank the three anonymous reviewers who helped, through their remarks and suggestions, to considerably improve this article. During this work, Felix Cheysson was a Ph.D. student of UMR MIA-Paris, Université Paris-Saclay, AgroParisTech, INRAE; Epidemiology and Modeling of bacterial Evasion to Antibacterials Unit (EMEA), Institut Pasteur and Centre de recherche en Epidémiologie et Santé des Populations (CESP), Université Paris-Saclay, UVSQ, Inserm.
Citation
Felix Cheysson. Gabriel Lang. "Spectral estimation of Hawkes processes from count data." Ann. Statist. 50 (3) 1722 - 1746, June 2022. https://doi.org/10.1214/22-AOS2173
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