  The multivariate Hawkes process is a past-dependent point process used to
model the relationship of event occurrences between different
phenomena.Although the Hawkes process was originally introduced to describe
excitation effects, which means that one event increases the chances of another
occurring, there has been a growing interest in modelling the opposite effect,
known as inhibition.In this paper, we focus on how to infer the parameters of a
multidimensional exponential Hawkes process with both excitation and inhibition
effects. Our first result is to prove the identifiability of this model under a
few sufficient assumptions. Then we propose a maximum likelihood approach to
estimate the interaction functions, which is, to the best of our knowledge, the
first exact inference procedure in the frequentist framework.Our method
includes a variable selection step in order to recover the support of
interactions and therefore to infer the connectivity graph.A benefit of our
method is to provide an explicit computation of the log-likelihood, which
enables in addition to perform a goodness-of-fit test for assessing the quality
of estimations.We compare our method to standard approaches, which were
developed in the linear framework and are not specifically designed for
handling inhibiting effects.We show that the proposed estimator performs better
on synthetic data than alternative approaches. We also illustrate the
application of our procedure to a neuronal activity dataset, which highlights
the presence of both exciting and inhibiting effects between neurons.
