Statistical inference for a partially observed interacting system of Hawkes processes

Chenguang Liu

Stochastic Processes and their Applications, 2020, vol. 130, issue 9, 5636-5694

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.

Keywords: Multivariate Hawkes processes; Point processes; Statistical inference; Interaction graph; Stochastic interacting particles; Mean field limit (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spa.2020.04.003

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