Since both model () and () have the same vector form (), we consider estimation of these two models simultaneously.
as the minimizers of the following
penalized weighted least squares
The following iterative procedure is used to solve ().
Algorithm Estimate , and iteratively using the following two steps:
(a) Given the current estimates of and , update ;
(b) Given the current estimates of , update and .
In step (a), if is linear in , then model () is a SSR model. Thus the solutions have the form (). After certain transformations, we can call ssr to update . If is nonlinear in , then model () is a NNR model. Thus the closed form of solutions do not exist. We can approximate the solutions as in NNR models. After certain transformations, we can call nnr to update .
In step (b), () is a regular parametric nonlinear regression model when is fixed. Thus we can update and using the S function gnls. We implemented the algorithm above by calling ssr/nnr and gnls alternately.
Conditional on , one can construct Bayesian confidence intervals as before. Adjustments need to be made to account for the loss of the degrees of freedom when is estimated. See Ke and Wang (2002) for more detailed discussions.