Statistical inference versus mean field limit for Hawkes processes
Résumé
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.