Parameter Estimation in Differential Equation Based Models

There is a long history for differential equations to be utilized to model dynamic pro- cesses in many disciplines such as physics, engineering, computer science, Finance, Biology, etc.. Most original efforts have been devoted to simulating the dynamic process for given parame- ters that characterize the differential equations. In recent years, more and more attention has been given by scientists, especially statisticians who considered the problem inversely, that is, using the experimental data to recover the values of parameters that specifically describe the experimental process trajectory.In this dissertation, we introduced a general Integro-ordinary differential equation that describes a reaction diffusion process for the tip growth of pollen tubes and proposed a constrained nonlinear mixed effects model for the dynamic response. Accordingly, we developed two estimation procedures to estimate this model, Constrained Method of Moments (CMM) and Constrained Restricted Maximum Likelihood (CREML). The advantages and disadvantages of the two procedures were investigated. Simulation studies and a real data analysis demonstrated that both estimation procedures could provide accurate estimates for the parameters.As an extension, the parameter estimation for Integro-partial differential equation of multiple dimension would also be discussed