Statistical inference versus mean field limit for Hawkes processes

Abstract : We consider a population of N N individuals, of which we observe the number of actions until time t t . For each couple of individuals (i,j) i j , j j may or not influence i i , which we model by i.i.d. Bernoulli(p) p -random variables, for some unknown parameter p∈(0,1] p 0 1 . Each individual acts autonomously at some unknown rate μ>0 μ 0 and acts by mimetism at some rate proportional to the sum of some function φ φ of the ages of the actions of the individuals which influence him. The function φ φ is unknown but assumed, roughly, to be decreasing and with fast decay. The goal of this paper is to estimate p p , which is the main characteristic of the graph of interactions, in the asymptotic N→∞ N ∞ , t→∞ t ∞ . The main issue is that the mean field limit (as N→∞ N ∞ ) of this model is unidentifiable, in that it only depends on the parameters μ μ and pφ p φ . Fortunately, this mean field limit is not valid for large times. We distinguish the subcritical case, where, roughly, the mean number mt m t of actions per individual increases linearly and the supercritical case, where mt m t increases exponentially. Although the nuisance parameter φ φ is non-parametric, we are able, in both cases, to estimate p p without estimating φ φ in a nonparametric way, with a precision of order N−1/2+N1/2m−1t 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.
Type de document :
Article dans une revue
Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2016, 10 (1), pp.1223-1295
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https://hal.archives-ouvertes.fr/hal-01367752
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Soumis le : vendredi 16 septembre 2016 - 15:56:44
Dernière modification le : mercredi 21 mars 2018 - 18:56:49

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• HAL Id : hal-01367752, version 1

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Nicolas Fournier, Sylvain Delattre. Statistical inference versus mean field limit for Hawkes processes. Electronic journal of statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2016, 10 (1), pp.1223-1295. 〈hal-01367752〉

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