Empirical Likelihood Inference For Ordinary Differential Equation Models With Error-prone State Variables

Differential equation models are widely used in many scientific fields, including engineering, physics and biomedical sciences. The so-called "forward problem", the problem of simulations and predictions of state variable for given parameter values in the differential equation models, has been extensively studied by mathematicians, physicists, engineers and other scientists. However, the "inverse problem", the prob-lem of parameter estimation based on the measurement of output variables, has not been well explored using modern statistical methods, although some least squares-based ap-proaches have been proposed and studied. In this paper we propose parameter estima-tion methods for ordinary differential equation models based on the local smoothing approach and empirical likelihood principle under a framework of measurement error in regression models. The main results of this paper are as follows:1. In the first chapter, we give a brief description of empirical likelihood approach and local smoothing approach. The background and present conditions for statistical diagnostics are introduced.2. In the second chapter, we study systematically the parametric empirical like-lihood diagnostics. So-called "forward problem" is the problem of simulations and predictions of state variables for given parameter values in differential equation mod-els. However, the "inverse problem" is the problem of parametric diagnostics based on the measurement of output variables. In this paper, we study the parametric em-pirical likelihood diagnostics for "inverse problem". Firstly, we can't observe directly state variable, but we can observe its surrogate. Here we assume an additive measure-ment error model to relate the state variable to the surrogate. So we need to propose local smoothing estimation to estimate the state variable and its derivative. Secondly, we study the parametric empirical likelihood diagnostics in parametric ordinary differ-ential equation models. We construct profile empirical likelihood ratio and study its asymptotic results. Then we construct confidence interval for parametric component.3. In the third chapter, we extend parametric models to the semi-parametric models. We study systematically parametric empirical likelihood diagnostics in semi-parametric models. We are interested in parametric component of semi-parametric models, but we need to estimate non-parametric component. We propose local smooth-ing approach to estimate non-parametric component. We substitute a smoothing esti-mate of the non-parametric component in semi-parametric differential equation models to study statistical diagnostics for parametric component. We propose profile likelihood ratio based on empirical likelihood. Then we also study the asymptotic distribution for empirical likelihood ratio and construct the confidence interval for parametric compo-nent.4. In the fourth chapter, we carry out some simulations to compare empirical likelihood approach with pseudo least-square approach.5. In the fifth chapter, we prove the main results proposed in the previous chapters.