Likelihood based inference for the multivariate renewal Hawkes process

The recent introduction of the renewal Hawkes (RHawkes) process has extended the modeling capabilities of the classical Hawkes self-exciting process by allowing the immigrant arrival times to follow a general renewal process rather than a homogeneous Poisson process. A multivariate extension to the RHawkes process will be proposed, which allows different event types to interact with self- and cross-excitation effects, termed the multivariate renewal Hawkes (MRHawkes) process model. A recursive algorithm is developed to directly compute the likelihood of the model, which forms the basis of statistical inference. A modified algorithm for likelihood evaluation is also proposed which reduces computational time. The likelihood evaluation algorithm also implies a procedure to assess the goodness-of-fit of the temporal patterns of the events and distribution of the event types by computing independent and uniform residuals. The plug-in predictive density function for the next event time and methods to make future predictions using simulations are presented. Simulation studies will show that the likelihood evaluation algorithms and the prediction procedures are performing as expected. To illustrate the proposed methodology, data on earthquakes arising in two Pacific island countries Fiji and Vanuatu and trade-through data for the stock BNP Paribas on the Euronext Paris stock exchange are analyzed.