Statistical inference for a partially observed interacting system of Hawkes processes.

Summary We observe the actions of a K subsample of N individuals, during a time interval of length t>0, for some large K≤N. We model the individuals' relationships by i.i.d. Bernoulli (p) random variables, where p∈(0,1] is an unknown parameter. The action rate of each individual depends on an unknown parameter μ>0 and on the sum of some function ϕ of the ages of the actions of the individuals that influence it. The function ϕ is unknown but we assume that it decays quickly. The goal of this thesis is to estimate the parameter p, which is the main feature of the interaction graph, in the asymptotic where population size N→∞, the observed population size K→∞, and in a long time t→∞. Let mt be the average number of actions per individual up to time t, which depends on all model parameters. In the subcritical case, where mt increases linearly, we construct an estimator of p with convergence rate 1K√+NmtK√+NKmt√. In the supercritical case, where mt increases exponentially fast, we construct an estimator of p with convergence rate 1K√+NmtK√. In a second step, we study the asymptotic normality of these estimators. In the subcritical case, the work is very technical but quite general, and we are led to study three possible regimes, depending on the dominant term in 1K√+NmtK√+NKmt√ at 0. In the supercritical case, we unfortunately assume some additional conditions and consider only one of the two possible regimes.