R help - Apr 2005 - gnlr3 location parameter

Hi list,

my previous question was obviously too basic to deserve an answer - 
apologies for that. I'm learning, things can only get better :-)

My current problem is with fitting a generalized gamma distribution with 
an additional "shift" parameter, that represents a shift of the 
distribution along the X axis.

The gnlr3 function (in Jim Lindsey's GNLM package) fits this 
distribution in this form:

f(y) = fy^(f-1)/((m/s)^(fs) Gamma(s)) y^(f(s-1)) exp(-(y s/m)^f)
(1)

I would like to include a fourth parameter, say u, like this:

f(y) = fy^(f-1)/((m/s)^(fs) Gamma(s)) (y-u)^(f(s-1)) exp(-((y-u) s/m)^f)
(2)

My best idea so far is to iteratively fit expression (1), each time 
shifting the data with an amount u. Plotting the maximum likelihood of 
the fit against u should give me an idea of where the optimum value for 
u is. Of course, this procedure will take quite some time, and will not 
be very straightforward since the generalized gamma shows convergence 
problems without good initial estimates...

Any suggestions for a better approach?

Thanks in advance,
Arnout

I believe what Jim Lindsey's code does is to directly maximize the 
log-likelihood.  Why not write down the log-likelihood for your problem 
and maximize it? You may be able to use the functions in package stats4 to 
provide a structure, or you can copy examples like fitdistr and polr in 
MASS.

Just be a little careful: you have omitted the ranges on your expressions, 
but is it not y > 0 for (1) and y > u for (2, corrected)?  If so you will 
need to use bound-constrained optimization and worry about having a 
non-standard inference problem.

Prof Lindsey chooses not to submit his code to CRAN (nor even keep it that 
at a stable URL).  As a result, few people here know about his packages 
and you would do better to ask him directly for support.

On Fri, 15 Apr 2005, Arnout Standaert wrote:
> Hi list,
>
> my previous question was obviously too basic to deserve an answer -
apologies
> for that. I'm learning, things can only get better :-)
>
> My current problem is with fitting a generalized gamma distribution with an
> additional "shift" parameter, that represents a shift of the
distribution
> along the X axis.
>
> The gnlr3 function (in Jim Lindsey's GNLM package) fits this
distribution in
> this form:
>
> f(y) = fy^(f-1)/((m/s)^(fs) Gamma(s)) y^(f(s-1)) exp(-(y s/m)^f)
> (1)
>
> I would like to include a fourth parameter, say u, like this:
>
> f(y) = fy^(f-1)/((m/s)^(fs) Gamma(s)) (y-u)^(f(s-1)) exp(-((y-u) s/m)^f)
> (2)
Is that right?  Did you mean (y-u) near the front?
> My best idea so far is to iteratively fit expression (1), each time
shifting
> the data with an amount u. Plotting the maximum likelihood of the fit
against
> u should give me an idea of where the optimum value for u is. Of course,
this
> procedure will take quite some time, and will not be very straightforward 
> since the generalized gamma shows convergence problems without good initial
> estimates...
>
> Any suggestions for a better approach?
>
> Thanks in advance,
> Arnout
>
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>
-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
1 South Parks Road,                     +44 1865 272866 (PA)
Oxford OX1 3TG, UK                Fax:  +44 1865 272595