Estimating parameters of continuous-time linear birth-death-immigration
processes, observed discretely at unevenly spaced time points, is a recurring
theme in statistical analyses of population dynamics. Viewing this task as a
missing data problem, we develop two novel expectation-maximization (EM)
algorithms. When the rate of immigration is either zero or proportional to the
birth rate, we use Kendall's generating function method to reduce the E-step of
the EM algorithm, as well as calculation of the Fisher information, to one
dimensional integration. This reduction results in simple and fast
implementation of the EM algorithm. To tackle the non-constrained immigration
rate, we extend a direct sampler for finite-state Markov chains and use this
sampling procedure to develop a Monte Carlo EM algorithm. We test our
algorithms on simulated data and then use our new methods to explore birth and
death rates of a transposable element in the genome of Mycobacterium
tuberculosis, the causative agent of tuberculosis.