Parameter estimation of the intensity process of self-exciting point processes using the EM algorithm

This paper presents a method of estimating the parameters of intensity processes in the self-exciting point process (SEPP) with the expectation-maximization (EM) algorithm. In the present paper, the case is considered where the intensity process of SEPPs is dependent only on the latest occurrence, i.e., one-memory SEPPs, as well as where the impulse response function characterizing the intensity process is parameterized as a single exponential function having a constant coefficient that takes a positive or negative value, i.e., making it possible to model a self "-exciting" or "-inhibiting" point process. Then, an explicit formula is derived for estimating the parameters specifying the intensity process on the basis of the EM algorithm, which in this instance gives the maximum likelihood (ML) estimates without solving nonlinear optimization problems. In practical computations, the parameters of interest can be estimated from the histogram of time intervals between point events. Monte Carlo simulations illustrate the validity of the derived estimation formulas and procedures.