  Birth-and-death processes are widely used to model the development of
biological populations. Although they are relatively simple models, their
parameters can be challenging to estimate, because the likelihood can become
numerically unstable when data arise from the most common sampling schemes,
such as annual population censuses. Simple estimators may be based on an
embedded Galton-Watson process, but this presupposes that the observation times
are equi-spaced. We estimate the birth, death, and growth rates of a linear
birth-and-death process whose population size is periodically observed via an
embedded Galton-Watson process, and by maximizing a saddlepoint approximation
to the likelihood. We show that a Gaussian approximation to the
saddlepoint-based likelihood connects the two approaches, we establish
consistency and asymptotic normality of quasi-likelihood estimators, compare
our estimators on some numerical examples, and apply our results to census data
for two endangered bird populations and the H1N1 influenza pandemic.
