What makes your model non-linear? I don't know your level of statistical knowledge. I have seen some people call a model with quadratic terms and interactions non-linear. As far and linear regression modeling goes, these are not non-linear. 

Leonard A Smith, thank you!

It would be great if you could give me the full reference of the paper you mention. I am trying to estimate the model parameters for a bio-process. I have worked with bio-hydrogen, but I am in general interested in bio-fuels. The model is a stiff system which consists of nonlinear ODEs. I could say that my final aim is to predict the dynamic performance of the process, use this model (with optimal parameters) for (economic) model predictive control.

Thanks once again.

Andrew Ekstrom I am not a specialist in statistics, nor do I have an advanced diploma in the subject, however I consider myself well rounded. The models I work with have exponentials as well as other non quadratic or linear terms (with respect to the parameters).

Dear Ehecatl,  How are you proceeding?  Are you using a curve-fitting package or are

you coding your own program, perhaps using a provided library of numerical routines?

Although the MLE and WLS approaches are not identical when constraints

are active or when your model is non-linear in the parmeters, as

it almost certainly is,  the more important thing is to explore your parameter space

and come up with a good fit that has meaning in the context you are fitting for.

Take a look at the story on fitting an IL-13 binding model and note the

numerous issues involved in such modeling. 

(see:  www.civilized.com/pdffiles/IL13.pdf)

Dear Gary Knott,

Thanks for your reply! I am coding my own program in Python using numerical routines as you mentioned. Thank you for the document you provided, it was quite useful.

Hello. I would say that WLS is the same as MLE in the linear estimation case when the weighting matrix is the noise covariance and the errors are zero-mean Gaussian. For nonlinear estimation there is no guarantee of convergence and, as Leonard Smith points out, the distributions will not be Gaussian in general, so the least squares metric that you get from a Gaussian distributed parameter vector will not apply.

If you add in constraints on the parameters you are in a totally different space. You can't even use Gauss-Newton or iterated least squares (see link below). You will need a constrained optimisation technique. The easiest way might be to add a penalty function and then treat as an unconstrained nonlinear estimation problem.

Article Analysis of a Nonlinear Least Squares Procedure Used in Glob...

Hello Graham W Pulford, thanks for your reply! Until now I have been using nonlinear constrained optimization, I generally use either interior point or sequential quadratic methods. A penalty function might be a bit difficult as I have many constraints, however the approach you sent me seems very interesting. I will definitely give it a go.

Thanks again!

Those interested at this discussion topic may possibly want to check this somewhat related RG thread: https://www.researchgate.net/post/anybody_help_me_use_LMS_algorithm_to_extract_the_coefficients_on_nonlinear_equation