Maximum likelihood estimation of the parameters $\lambda$ and $\mu$ of a simple (linear) birth-and-death process observed continuously over a fixed time interval is studied. Asymptotic distributions for large initial populations and for large periods of observation are derived and some nonstandard results appear. The result problem of estimation from the discrete skeleton of the process is also discussed.
"Maximum Likelihood Estimation in the Birth-and-Death Process." Ann. Statist. 3 (2) 363 - 372, March, 1975. https://doi.org/10.1214/aos/1176343062