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Statistical inference for a partially observed interacting system of Hawkes processes

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  • Liu, Chenguang

Abstract

We observe the actions of a K sub-sample of N individuals up to time t for some large K≤N. We model the relationships of individuals by i.i.d. Bernoulli(p)-random variables, where p∈(0,1] is an unknown parameter. The rate of action of each individual depends on some unknown parameter μ>0 and on the sum of some function ϕ of the ages of the actions of the individuals which influence him. The function ϕ is unknown but we assume it rapidly decays. The aim of this paper is to estimate the parameter p asymptotically as N→∞, K→∞, and t→∞. Let mt be the average number of actions per individual up to time t. In the subcritical case, where mt increases linearly, we build an estimator of p with the rate of convergence 1K+NmtK+NKmt. In the supercritical case, where mt increases exponentially fast, we build an estimator of p with the rate of convergence 1K+NmtK.

Suggested Citation

  • Liu, Chenguang, 2020. "Statistical inference for a partially observed interacting system of Hawkes processes," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5636-5694.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:9:p:5636-5694
    DOI: 10.1016/j.spa.2020.04.003
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    References listed on IDEAS

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    1. Jakob Gulddahl Rasmussen, 2013. "Bayesian Inference for Hawkes Processes," Methodology and Computing in Applied Probability, Springer, vol. 15(3), pages 623-642, September.
    2. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    3. Mohler, G. O. & Short, M. B. & Brantingham, P. J. & Schoenberg, F. P. & Tita, G. E., 2011. "Self-Exciting Point Process Modeling of Crime," Journal of the American Statistical Association, American Statistical Association, vol. 106(493), pages 100-108.
    4. Luc Bauwens & Nikolaus Hautsch, 2009. "Modelling Financial High Frequency Data Using Point Processes," Springer Books, in: Thomas Mikosch & Jens-Peter Kreiß & Richard A. Davis & Torben Gustav Andersen (ed.), Handbook of Financial Time Series, chapter 41, pages 953-979, Springer.
    5. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
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