  We introduce the Hawkes process with renewal immigration and make its
statistical estimation possible with two Expectation Maximization (EM)
algorithms. The standard Hawkes process introduces immigrant points via a
Poisson process, and each immigrant has a subsequent cluster of associated
offspring of multiple generations. We generalize the immigration to come from a
Renewal process; introducing dependence between neighbouring clusters, and
allowing for over/under dispersion in cluster locations. This complicates
evaluation of the likelihood since one needs to know which subset of the
observed points are immigrants. Two EM algorithms enable estimation here: The
first is an extension of an existing algorithm that treats the entire branching
structure - which points are immigrants, and which point is the parent of each
offspring - as missing data. The second considers only if a point is an
immigrant or not as missing data and can be implemented with linear time
complexity. Both algorithms are found to be consistent in simulation studies.
Further, we show that misspecifying the immigration process introduces
signficant bias into model estimation-- especially the branching ratio, which
quantifies the strength of self excitation. Thus, this extended model provides
a valuable alternative model in practice.
