Entropy-based estimation of the birth-death ratio

A population is modeled by a birth-death process in a finite state space. Its stationary distribution is indexed by its birth-death ratio. A sample of values taken by the population size has an elastic sample mean (mean of the observations), an additional sample mean (mean of the logarithms of the observations transformed by a given function), and a synchronizing sample mean (combination of the previous means). When the last two means are zero, then, by definition, information is linear in population size. This is only the case when the population size is geometrically distributed. Equalizing the entropy of a distribution to the entropy calculated on any sample involves the three sample means and allows for estimating the birth-death ratio. Only in the case of information linear in population size, this procedure reduces to maximum likelihood estimation, which involves only the elastic sample mean. The procedure is demonstrated on information that is no longer linear in population size, such as a binomial distribution of population size, where the last two means are not zero, but just equal, and a Pascal distribution and a Poisson distribution, where the last two means are neither zero nor equal.