## The Annals of Statistics

- Ann. Statist.
- Volume 38, Number 1 (2010), 435-481.

### Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations

Xin Qi and Hongyu Zhao

#### Abstract

Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop efficient estimation procedures for these parameters. Among the proposed methods in the literature, the generalized profiling estimation method developed by Ramsay and colleagues is particularly promising for its computational efficiency and good performance. In this approach, the ODE solution is approximated with a linear combination of basis functions. The coefficients of the basis functions are estimated by a penalized smoothing procedure with an ODE-defined penalty. However, the statistical properties of this procedure are not known. In this paper, we first give an upper bound on the uniform norm of the difference between the true solutions and their approximations. Then we use this bound to prove the consistency and asymptotic normality of this estimation procedure. We show that the asymptotic covariance matrix is the same as that of the maximum likelihood estimation. Therefore, this procedure is asymptotically efficient. For a fixed sample and fixed basis functions, we study the limiting behavior of the approximation when the smoothing parameter tends to infinity. We propose an algorithm to choose the smoothing parameters and a method to compute the deviation of the spline approximation from solution without solving the ODEs.

#### Article information

**Source**

Ann. Statist. Volume 38, Number 1 (2010), 435-481.

**Dates**

First available in Project Euclid: 31 December 2009

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1262271620

**Digital Object Identifier**

doi:10.1214/09-AOS724

**Zentralblatt MATH identifier**

1181.62156

**Mathematical Reviews number (MathSciNet)**

MR2589327

**Subjects**

Primary: 62F12: Asymptotic properties of estimators

Secondary: 65L05: Initial value problems 65D07: Splines

**Keywords**

Ordinary differential equations parameters estimation profiling procedure consistency asymptotic normality

#### Citation

Qi, Xin; Zhao, Hongyu. Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations. The Annals of Statistics 38 (2010), no. 1, 435--481. doi:10.1214/09-AOS724. http://projecteuclid.org/euclid.aos/1262271620.