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

We estimate
and
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.