A Tracking Approach to Parameter Estimation in Linear Ordinary Differential Equations

Ordinary Differential Equations are widespread tools to model chemical, physical, bio-logical process but they usually rely on parameters which are of critical importance in terms of dynamic and need to be estimated directly from the data. Classical statistical approaches (nonlinear least squares, maximum likelihood estimator) can give unsatis-factory results because of computational diculties and ill-posedness of the statistical problem. New estimation methods that use some nonparametric devices have been proposed to circumvent these issues. We present a new estimator that shares prop-erties with Two-Step estimator and Generalized Smoothing (introduced by Ramsay et al. [34]). We introduce a perturbed model and we use optimal control theory for constructing a criterion that aims at minimizing the discrepancy with data and the model. Here, we focus on the case of linear Ordinary Dierential Equations as our criterion has a closed-form expression that permits a detailed analysis. Our approach avoids the use of a nonparametric estimator of the derivative, which is one of the main cause of inaccuracy in Two-Step estimators. Moreover, we take into account model discrepancy and our estimator is more robust to model misspecication than classical methods. The discrepancy with the parametric ODE model correspond to the mini-mum perturbation (or control) to apply to the initial model. Its qualitative analysis can be informative for misspecication diagnosis. In the case of well-specied model, we show the consistency of our estimator and that we reach the parametric √ n− rate when regression splines are used in the rst step