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2016 Statistical inference versus mean field limit for Hawkes processes
Sylvain Delattre, Nicolas Fournier
Electron. J. Statist. 10(1): 1223-1295 (2016). DOI: 10.1214/16-EJS1142


We consider a population of $N$ individuals, of which we observe the number of actions until time $t$. For each couple of individuals $(i,j)$, $j$ may or not influence $i$, which we model by i.i.d. Bernoulli$(p)$-random variables, for some unknown parameter $p\in(0,1]$. Each individual acts autonomously at some unknown rate $\mu>0$ and acts by mimetism at some rate proportional to the sum of some function $\varphi$ of the ages of the actions of the individuals which influence him. The function $\varphi$ is unknown but assumed, roughly, to be decreasing and with fast decay. The goal of this paper is to estimate $p$, which is the main characteristic of the graph of interactions, in the asymptotic $N\to\infty$, $t\to \infty$. The main issue is that the mean field limit (as $N\to\infty$) of this model is unidentifiable, in that it only depends on the parameters $\mu$ and $p\varphi$. Fortunately, this mean field limit is not valid for large times. We distinguish the subcritical case, where, roughly, the mean number $m_{t}$ of actions per individual increases linearly and the supercritical case, where $m_{t}$ increases exponentially. Although the nuisance parameter $\varphi$ is non-parametric, we are able, in both cases, to estimate $p$ without estimating $\varphi $ in a nonparametric way, with a precision of order $N^{-1/2}+N^{1/2}m_{t}^{-1}$, up to some arbitrarily small loss. We explain, using a Gaussian toy model, the reason why this rate of convergence might be (almost) optimal.


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Sylvain Delattre. Nicolas Fournier. "Statistical inference versus mean field limit for Hawkes processes." Electron. J. Statist. 10 (1) 1223 - 1295, 2016.


Received: 1 September 2015; Published: 2016
First available in Project Euclid: 12 May 2016

zbMATH: 1343.62050
MathSciNet: MR3499526
Digital Object Identifier: 10.1214/16-EJS1142

Primary: 60J75, 60K35, 62M09

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society


Vol.10 • No. 1 • 2016
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