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Abstract
The class of self-exciting point process evolve within a self-excitation mechanism that allows past events to contribute to the arrival rate of future events. The significant contributions this thesis introduces are techniques to conduct efficient statistical inferences for the recently proposed renewal Hawkes self-exciting point processes. By employing a substantial modification to the baseline arrival rate of the Hawkes process, the renewal Hawkes process provides superior versatility. The additional flexibility afforded to the renewal Hawkes process occurs by defining the immigration process in terms of a general renewal process rather than a homogenous Poisson process. The renewal Hawkes process has the potential to widen the application domains of self-exciting processes significantly. However, it was initially asserted that likelihood evaluation of the process demands exponential computational time and therefore is practically impossible. As a consequence, two Expectation-Maximization (E-M) algorithms were developed to compute the maximum likelihood estimator (MLE), a bootstrap procedure to estimate the variance-covariance matrix of the MLE and a Monte Carlo approach to compute a goodness-of-fit test statistic.
Considering the fundamental role played by the likelihood function in statistical inferences, a practically feasible method for likelihood evaluation is highly desirable. This thesis develops algorithms to evaluate the likelihood of the renewal Hawkes process in quadratic time, a drastic improvement from the exponential time initially claimed. Simulations will demonstrate the superior performance of the resulting MLEs of the model relative to the E-M estimators. This thesis will also introduce computationally efficient procedures to calculate the Rosenblatt residuals of the process for goodness-of-fit assessment and a simple yet efficient procedure for future event predictions. Faster fitting methods, and linear time algorithms to fit the process are also discussed. The computational efficiency of the methods developed facilitates the application of these algorithms to multi-event and marked point process models with renewal immigration. As such, this thesis proposes two additional models termed the multivariate renewal Hawkes process and the mark renewal Hawkes process. The additional computational challenges that arise in these frameworks are solved herein.