Re: Constrained non linear regression using ML

[Michael Creel]

ct529 wrote:
> 
> Dear Friedrik, Jaroslav, Octave,
> 
> Yes, of course it depends on the error. But you can still build a 
> frequency distribution with the y_j. It was additional information on 
> the problem, but probably not very useful, apologies.
> 
> First of all, the {p1,....,pn} and hence the {k1,....,kn} can have a few 
> tenths of elements (that is n could be maybe 60 in the worst case). The 
> problem is that for such a case we use millions of observations ;). In 
> the case of the 40,000 observation it would be safe to suppose we would 
> use a max of 20.
> 
> At the moment I have a few of assumptions on the error:
> 
> 1) Assumption 1:
> 
> error is normally distributed with mean 0. In this case I can use NLS, 
> what do you think?
> 
> 2) Assumption 2:
> 
> error is normally distributed with mean 0 AFTER inverse transformation, 
> so you fit on x~ k0+k1*p1+ .... + kn*pn using NLS again. What do you
> think?
> 
> 3) Assumption 3:
> 
> error is beta distributed. I have no idea.
> 
> 4) Assumption 4:
> 
> error is beta distributed AFTER inverse transformation. I have no idea.
> 
> 5) Assumption 5:
> 
> error is distributed with a distribution in the exponential family .... 
> I have no clue.
> 
> 6) Assumption 6:
> 
> error is distributed with a distribution in the exponential family after 
> transformation .... I have no clue.
> 
> Assumption 3 to 6 are the most important, and I would like to be able to 
> build a very generic package that covers most of the important cases. I 
> would also like to be able to change the link function but that is for 
> much later.
> 
> What do you suggest?
> 
> 

Regarding the possibilities >2, the econometrics package in OF contains a
maximum likelihood routine for nonlinear models. There is an example
"mle_example.m" that shows how to fit a Poisson model. You could easily
adapt that to estimate a beta model. For parameters that must be
non.negative, you can parameterize k = exp(kstar) so that kstar is
unrestricted. Then you estimate the kstars. The delta method (first order
Taylor's series correction) can be used to recover the estimated covariance
of the restricted parmeters (k)  from the estimated covariance of the
unrestricted parameters (kstar). This approach does require that you specify
the distribution, beta or whatever.
Cheers, Michael

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