Statistical inference for a partially observed interacting system of Hawkes processes
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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 m_t be the average number of actions per individual up to time t. In the subcritical case, where m_t increases linearly, we build an estimator of p with the rate of convergence 1/√(K)+N/m_t√(K)+N/K√(m_t). In the supercritical case, where m_t increases exponentially fast, we build an estimator of p with the rate of convergence 1/√(K)+N/m_t√(K).
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