The ARMA Point Process and its Estimation

An autoregressive–moving-average (ARMA) point process model is introduced, which combines self-exciting and shot-noise cluster mechanisms, both useful in a variety of applications. The process is analogous to the ARMA for integer-valued time series, sharing methodological and mathematical similarities. A maximum likelihood estimation procedure, based on MCEM (Monte Carlo Expectation Maximization), is derived and studied. This approach conveniently allows for: (i) trends in immigration/background intensity, (ii) multiple parametric specifications of memory functions and mark distributions, as well as (iii) cases where marks and immigrants are not observed. As such, the ARMA point process provides a flexible framework to disentangle cluster mechanisms in continuously observed count data.