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Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters. (English) Zbl 1327.62120
Summary: Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their ’true’ value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters.
For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their \(\sqrt{n}\)-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches.

MSC:
62F12 Asymptotic properties of parametric estimators
62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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