From the paper: Construction of models is the most basic activity in science, lying at the heart of understanding, prediction and practical applications. Uncovering the dominant mechanisms governing the deterministic evolution of nonlinear dynamical systems is an essential component of this activity. These systems are known to exhibit a wide variety of behaviour, ranging from chaos to regular trains of discrete events. Various methods are employed to reduce the complexity of such problems. One important approach is to exploit the nonlinear interaction of quantities varying on different time- and length-scales. This often makes it possible to eliminate rapidly fluctuating variables and represent the evolution of important properties of the system by random processes. Identifying stochastic models governing the behaviour of these properties then becomes the important driver for data acquisition and analysis. In reality, measurements are always limited by practical considerations. Only finite datasets are available, and these may relate only indirectly to the processes of interest. Thus distinguishing different models may be difficult. We investigate a very simple version of this problem. We present a detailed study of two rather different stochastic population processes that generate identical first-order statistics and are monitored indirectly through the rate at which individuals migrate. We investigate the statistical and correlation properties of two stochastic population models that give rise to identical first-order probability densities. We assume that the processes are monitored indirectly through measurement of the rate at which individuals emigrate from the population. Formulae characterizing the integrated statistics of these counting processes are derived, and it is shown how they may be used to distinguish the population models.