  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.
