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

TL;DR

This paper analyzes the statistical properties of the generalized profiling estimation method for parameters in ODEs, proving its asymptotic efficiency and providing practical algorithms for parameter selection and approximation accuracy.

Contribution

It establishes the consistency, asymptotic normality, and efficiency of the generalized profiling estimation method for ODE parameters, and introduces algorithms for smoothing parameter selection.

Findings

The estimation method is asymptotically efficient with an asymptotic covariance matching MLE.

Provided bounds on the approximation error between true solutions and basis function approximations.

Developed algorithms for smoothing parameter choice and deviation computation without solving ODEs.

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.