Estimation of rates in population-age-dependent processes by means of test functions

This paper aims to develop practical applications of the model for the highly technical measure-valued populations developed by the authors in \cite{FanEtal20}. We consider the problem of estimation of parameters in the general age and population-dependent model, in which the individual birth and death rates depend not only on the age of the individual but also on the whole population composition. We derive new estimators of the rates based on the use of test functions in the functional Law of Large Numbers and Central Limit Theorem for populations with a large carrying capacity. We consider the rates to be simple functions, that take finitely many values both in age $x$ and measure $A$, which leads to systems of linear equations. The proposed method of using test functions for estimation is a radically new approach which can be applied to a wide range of models of dynamical systems.