Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters

TL;DR

This paper introduces an efficient estimation method for linear-in-parameters differential systems, providing identifiability conditions, optimal convergence rates, and an experimental design that enhances practical applicability.

Contribution

It presents a novel estimation approach that avoids numerical integration and derivative estimation, with proven $\

Findings

Establishes necessary and sufficient conditions for parameter identifiability.

Proposes an estimator with $\

Demonstrates $\

Abstract

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.