Estimating the Hawkes Process From a Discretely Observed Sample Path

Estimating the Hawkes process from a discretely observed sample path is challenging due to the intractability of the likelihood in such cases. To overcome this, we employ a state-space representation of the incomplete data problem and use the sequential Monte Carlo (SMC, aka particle filters) to approximate the likelihood function. The resulting estimator of the likelihood function is unbiased and, therefore, can be used along with the Metropolis-Hastings algorithm to construct Markov Chains to approximate the likelihood distribution and, more generally, the posterior distribution of model parameters. The performance of our methodology is assessed using simulation experiments and compared with other recently published methods. The proposed estimator exhibits a smaller mean square error compared to two benchmark estimators. Furthermore, an advantage of our method compared to existing methods is that confidence intervals for the parameters are readily computable. Finally, we apply the proposed estimator to the analysis of weekly count data on measles cases in Tokyo, Japan, and compare the results to one of the benchmark methods. The online supplementary materials contain a Julia package that implements our methodology, along with the technical proofs for two propositions. Supplementary materials for this article are available online.