Consistency of direct integral estimator for partially observed systems of ordinary differential equations linear in the parameters

Dynamic systems are ubiquitous in nature and are used to model many processes in biology, chemistry, physics, medicine, and engineering. In particular, systems of ordinary differential equations are commonly used for the mathematical modelling of the rate of change of dynamic processes. In many practical applications, the process can only be partially measured, a fact that renders estimation of parameters of the system extremely challenging. Recently, a 'direct integral estimator' for partially observed systems of ordinary differential equations was introduced. The practical performance of the integral estimator was demonstrated, but its theoretical properties were not derived. In this paper we use the sieve framework to prove that the estimator is consistent.