# Statistical inference for a partially observed interacting system of Hawkes processes

Abstract : We observe the actions of a $K$ sub-sample of $N$ individuals, during some time interval with length $t>0$, for some large $K\le N$. We model the relationships of individuals by i.i.d. Bernoulli($p$) random variables, where $p\in (0,1]$ is an unknown parameter. The rate of action of each individual depends on some unknown parameter $\mu> 0$ and on the sum of some function $\phi$ of the ages of the actions of the individuals which influence him. The function $\phi$ is unknown but we assume it rapidly decays. The aim of this thesis is to estimate the parameter $p$, which is the main characteristic of the interaction graph, in the asymptotic where the population size $N\to \infty$, the observed population size $K\to \infty$, and in large time $t\to \infty$. Let $m_t$ be the average number of actions per individual up to time $t$, which depends on all the parameters of the model. In the subcritical case, where $m_t$ increases linearly, we build an estimator of $p$ with the rate of convergence $\frac{1}{\sqrt{K}}+\frac{N} m_t\sqrt{K}}+\frac{N}{K\sqrt{m_t}}$. In the supercritical case, where $m_{t}$ increases exponentially fast, we build an estimator of $p$ with the rate of convergence $\frac{1}{\sqrt{K}}+\frac{N}{m_{t}\sqrt{K}}$. In a second time, we study the asymptotic normality of those estimators. In the subcritical case, the work is very technical but rather general, and we are led to study three possible regimes, depending on the dominating term in $\frac{1}{\sqrt{K}}+\frac{N}{m_t\sqrt{K}}+\frac{N}{K\sqrt{m_t}} \to 0$. In the supercritical case, we, unfortunately, suppose some additional conditions and consider only one of the two possible regimes.
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Submitted on : Tuesday, February 11, 2020 - 4:48:28 PM
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Chenguang Liu. Statistical inference for a partially observed interacting system of Hawkes processes. Statistics [math.ST]. Sorbonne Université, 2019. English. ⟨tel-02474901⟩

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