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Ordinary differential equations (ODEs) are commonly used to model dynamicbehavior of a system. Because many parameters are unknown and have to beestimated from the observed data, there is growing interest in statistics todevelop efficient estimation procedures for these parameters. Among theproposed methods in the literature, the generalized profiling estimation methoddeveloped by Ramsay and colleagues is particularly promising for itscomputational efficiency and good performance. In this approach, the ODEsolution is approximated with a linear combination of basis functions. Thecoefficients of the basis functions are estimated by a penalized smoothingprocedure with an ODE-defined penalty. However, the statistical properties ofthis procedure are not known. In this paper, we first give an upper bound onthe uniform norm of the difference between the true solutions and theirapproximations. Then we use this bound to prove the consistency and asymptoticnormality of this estimation procedure. We show that the asymptotic covariancematrix is the same as that of the maximum likelihood estimation. Therefore,this procedure is asymptotically efficient. For a fixed sample and fixed basisfunctions, we study the limiting behavior of the approximation when thesmoothing parameter tends to infinity. We propose an algorithm to choose thesmoothing parameters and a method to compute the deviation of the splineapproximation from solution without solving the ODEs.

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