Direct Likelihood Evaluation for the Renewal Hawkes Process
针对更新霍克斯过程似然计算需指数时间的问题，提出二次时间算法，通过模拟和真实数据（地震学、金融）证明其优于EM估计，并提供拟合优度检验和预测方法。
An interesting extension of the widely applied Hawkes self-exiting point process, the renewal Hawkes (RHawkes) process, was recently proposed by Wheatley, Filimonov, and Sornette, which has the potential to significantly widen the application domains of the self-exciting point processes. However, they claimed that computation of the likelihood of the RHawkes process requires exponential time and therefore is practically impossible. They proposed two expectation–maximization (EM) type algorithms to compute the maximum likelihood estimator (MLE) of the model parameters. Because of the fundamental role of likelihood in statistical inference, a practically feasible method for likelihood evaluation is highly desirable. In this article, we provide an algorithm that evaluates the likelihood of the RHawkes process in quadratic time, a drastic improvement from the exponential time claimed by Wheatley, Filimonov, and Sornette. We demonstrate the superior performance of the resulting MLEs of the model relative to the EM estimators through simulations. We also present a computationally efficient procedure to calculate the Rosenblatt residuals of the process for goodness-of-fit assessment, and a simple yet efficient procedure for future event prediction. The proposed methodologies were applied on real data from seismology and finance. An R package implementing the proposed methodologies is included in the supplementary materials.