Abstract: Model checking for parametric ordinary differential equations is a necessary step when checking whether the assumed models are plausible. In this paper, we first introduce a trajectory matching-based test for the whole model, which can also easily be applied to check partially observed systems. Then, we provide two tests to identify which component function is modeled incorrectly. The first is based on integral matching, and the second is based on gradient matching, with bias correction achieved using data splitting. We investigate their asymptotic properties under the null, global, and local alternative hypotheses. Because there are no results for relevant parameter estimations for alternative models in the literature, we also investigate the asymptotic properties of the nonlinear least squares estimation and the two-step estimation under both the null and the alternatives. To examine the performance of the tests, we conduct several numerical simulations and an analysis using a real-data example on immune cell kinetics and trafficking for influenza infection.

Key words and phrases: Local smoothing test, model checking, ordinary differential equations.