## Electronic Journal of Statistics

### Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters

#### 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.

#### Article information

Source
Electron. J. Statist., Volume 9, Number 2 (2015), 1939-1973.

Dates
First available in Project Euclid: 27 August 2015

https://projecteuclid.org/euclid.ejs/1440680332

Digital Object Identifier
doi:10.1214/15-EJS1053

Mathematical Reviews number (MathSciNet)
MR3391125

Zentralblatt MATH identifier
1327.62120

#### Citation

Dattner, Itai; Klaassen, Chris A. J. Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters. Electron. J. Statist. 9 (2015), no. 2, 1939--1973. doi:10.1214/15-EJS1053. https://projecteuclid.org/euclid.ejs/1440680332

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