Numerical discretization-based kernel type estimation methods for ordinary differential equation models

We consider the problem of parameter estimation in both linear and nonlinear ordinary differential equation (ODE) models. Nonlinear ODE models are widely used in applications. But their analytic solutions are usually not available. Thus regular methods usually depend on repetitive use of numerical solutions which bring huge computational cost. We proposed a new two-stage approach which includes a smoothing method (kernel smoothing or local polynomial fitting) in the first stage, and a numerical discretization method (Eulers discretization method, the trapezoidal discretization method, or the Runge-Kutta discretization method) in the second stage. Through numerical simulations, we find the proposed method gains a proper balance between estimation accuracy and computational cost. Asymptotic properties are also presented, which show the consistency and asymptotic normality of estimators under some mild conditions. The proposed method is compared to existing methods in term of accuracy and computational cost. The simulation results show that the estimators with local linear smoothing in the first stage and trapezoidal discretization in the second stage have the lowest average relative errors. We apply the proposed method to HIV dynamics data to illustrate the practicability of the estimator.