June 2022 Spectral estimation of Hawkes processes from count data
Felix Cheysson, Gabriel Lang
Author Affiliations +
Ann. Statist. 50(3): 1722-1746 (June 2022). DOI: 10.1214/22-AOS2173


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.


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.


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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


Received: 1 March 2020; Revised: 1 October 2021; Published: June 2022
First available in Project Euclid: 16 June 2022

MathSciNet: MR4441138
zbMATH: 1489.60082
Digital Object Identifier: 10.1214/22-AOS2173

Primary: 60G55
Secondary: 37A25 , 62M15

Keywords: count data , Hawkes process , spectral estimation , Strong mixing , Whittle method

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 3 • June 2022
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