Estimation

Since both model () and () have the same vector form (), we consider estimation of these two models simultaneously.

We estimate $\mbox{\boldmath$\phi$}$ and $\mbox{\boldmath$f$}$ as the minimizers of the following penalized weighted least squares

$$(\mbox{\boldmath$y$}-\mbox{\boldmath$\eta$}(\mbox{\boldmath$\phi$... ...um_{j=1}^q \sum_{k=1}^{p_j} \theta_{jk}^{-1} \vert\vert P_{jk}f_j\vert\vert^2 .$$ (39)

The following iterative procedure is used to solve ().

Algorithm Estimate $\mbox{\boldmath$f$}$, $\mbox{\boldmath$\phi$}$ and $\mbox{\boldmath$\tau$}$ iteratively using the following two steps:

(a) Given the current estimates of $\mbox{\boldmath$\phi$}$ and $\mbox{\boldmath$\tau$}$, update $\mbox{\boldmath$f$}$;

(b) Given the current estimates of $\mbox{\boldmath$f$}$, update $\mbox{\boldmath$\phi$}$ and $\mbox{\boldmath$\tau$}$.

In step (a), if $\eta$ is linear in $\mbox{\boldmath$f$}$, then model () is a SSR model. Thus the solutions have the form (). After certain transformations, we can call ssr to update $\mbox{\boldmath$f$}$. If $\eta$ is nonlinear in $\mbox{\boldmath$f$}$, 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 $\mbox{\boldmath$f$}$.

In step (b), () is a regular parametric nonlinear regression model when $\mbox{\boldmath$f$}$ is fixed. Thus we can update $\mbox{\boldmath$\phi$}$ and $\mbox{\boldmath$\tau$}$ using the S function gnls. We implemented the algorithm above by calling ssr/nnr and gnls alternately.

Conditional on $\mbox{\boldmath$\phi$}$, one can construct Bayesian confidence intervals as before. Adjustments need to be made to account for the loss of the degrees of freedom when $\mbox{\boldmath$\phi$}$ is estimated. See Ke and Wang (2002) for more detailed discussions.