Translator Disclaimer
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

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.

Citation

Download Citation

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

Information

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

Subjects:
Primary: 60J75, 60K35, 62M09

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

JOURNAL ARTICLE
73 PAGES


SHARE
Vol.10 • No. 1 • 2016
Back to Top