Moment estimator of the initial size of a branching process used to model polymerase chain reaction

Consider a binary-splitting population-size-dependent branching process for which the offspring from generation n+1, conditionally to the past of the process up to generation n, follow a Bernoulli distribution with probability of success depending on the population size at generation n according to a Michaelis-Menten type model. We aim at determining the initial population size of the process based on consecutive observations of the process. An estimator based on the method of moments is proposed and its behavior when considering a finite number of observations is given. The model presented here may be used to represent DNA amplification by PCR. In this molecular biology setting, the determination of the starting amount of DNA fragments presents important practical applications