ASYMPTOTIC EFFICIENCY AND FINITE-SAMPLE PROPERTIESOF THE GENERALIZED PROFILING ESTIMATION OFPARAMETERS IN ORDINARY DIFFERENTIAL EQUATIONS1

摘要

Ordinary differential equations (ODEs) are commonly used to model dy-namic behavior of a system. Because many parameters are unknown and haveto be estimated from the observed data, there is growing interest in statistics todevelop efficient estimation procedures for these parameters. Among the pro-posed methods in the literature, the generalized profiling estimation methoddeveloped by Ramsay and colleagues is particularly promising for its compu-tational efficiency and good performance. In this approach, the ODE solutionis approximated with a linear combination of basis functions. The coefficientsof the basis functions are estimated by a penalized smoothing procedure withan ODE-defined penalty. However, the statistical properties of this procedureare not known. In this paper, we first give an upper bound on the uniformnorm of the difference between the true solutions and their approximations.Then we use this bound to prove the consistency and asymptotic normality ofthis estimation procedure. We show that the asymptotic covariance matrix isthe same as that of the maximum likelihood estimation. Therefore, this proce-dure is asymptotically efficient. For a fixed sample and fixed basis functions,we study the limiting behavior of the approximation when the smoothing pa-rameter tends to infinity. We propose an algorithm to choose the smoothingparameters and a method to compute the deviation of the spline approxima-tion from solution without solving the ODEs.