Electronic Journal of Statistics

Statistical inference versus mean field limit for Hawkes processes

Sylvain Delattre and Nicolas Fournier

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Abstract

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.

Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 1223-1295.

Dates
Received: September 2015
First available in Project Euclid: 12 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1463068346

Digital Object Identifier
doi:10.1214/16-EJS1142

Mathematical Reviews number (MathSciNet)
MR3499526

Zentralblatt MATH identifier
1343.62050

Subjects
Primary: 62M09: Non-Markovian processes: estimation 60J75: Jump processes 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Multivariate Hawkes processes point processes statistical inference interaction graph stochastic interacting particle systems propagation of chaos mean field limit

Citation

Delattre, Sylvain; Fournier, Nicolas. Statistical inference versus mean field limit for Hawkes processes. Electron. J. Statist. 10 (2016), no. 1, 1223--1295. doi:10.1214/16-EJS1142. https://projecteuclid.org/euclid.ejs/1463068346


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