Maximum likelihood estimation for Hawkes processes with self-excitation or inhibition

Abstract

In this paper, we present a maximum likelihood method for estimating the parameters of a univariate Hawkes process with self-excitation or inhibition. Our work generalizes techniques and results that were restricted to the self-exciting scenario. The proposed estimator is implemented for the classical exponential kernel and we show that, in the inhibition context, our procedure provides more accurate estimations than current alternative approaches.

Inference of multivariate exponential Hawkes processes with inhibition and application to neuronal activity

“…Lastly, Bonnet et al (2021) presents a maximum likelihood estimation adapted to the univariate Hawkes process with inhibition and monotone kernel functions. The decisive contribution of this work is to give, for an exponential kernel h(t) = αe −βt (α ∈ R, β > 0), a closed-form expression of restart times, which are basically the instants at which the single conditional intensity becomes nonzero.…”

“…Yet, this study is limited to the univariate case. It has to be noted that a formalism similar to Bonnet et al (2021) but for multivariate Hawkes processes is mentioned in Deutsch and Ross (2022). However, the authors chose to put this framework aside and prefer to focus on non-parametric Bayesian estimation.…”

“…It remains now to determine when the underlying intensity λ i is negative. To do so, we leverage the work done in Bonnet et al (2021) and define the restart times in the multivariate framework, to be, for any k and i:…”

“…In this paper, we present a maximum likelihood estimation method for multivariate Hawkes processes with exponential kernel functions, that works for both exciting and inhibiting interactions, as modelled by Brémaud and Massoulié (1996); Chen et al (2017). This work builds upon the methodology for the univariate case, presented in Bonnet et al (2021), by focusing in the intervals where the intensity function is positive. We show that, under a weak assumption on the kernel functions, these intervals can be exactly determined.…”

“…In Bonnet et al (2021), the authors solved this challenge for univariate processes by remarking that the conditional intensity is monotone between two event times. Figure 1 illustrates that this is not necessarily true for multivariate processes (here, between T (2) and T (3) ).…”