error in ml stat

We have found an error in the maximum likelihood statistic in ISIS,
"ml".

Formula 7.51 (in the version of the manual dated 6/2/11) is incorrect.
Instead of ln Gamma[Mi+1], it should be ln Gamma[Ci+1]. It should also
have factor of 2 to make it distributed as chisq.  (Thanks to Herman
Marshall for providing details.  And for being the first user,
apparently, of "ml".)

I have tested this by defining user statistics as printed in the manual
(accidentally, but which did indicate that that is what is implemented
in ISIS) and with the correct formula (which gives good results).  I fit
a well-exposed emission line with a Gaussian plus polynomial, and I made
confidence contours in line center and area using "chisqr", "cash", and
"myml" stats, with similar results.  "myml" and "cash" gave identical
contours.

E.g.

> cat myml_stat.sl

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

require( "gsl" );

define myml_stat( y, fx, w )
{
    variable v = 2 * ( fx + lngamma( y + 1 ) - y * log( fx ) );
    return( v, sum( v ) );
}

define myml_report ( stat, npts, nvpars )
{
    variable s = sprintf ("  My ML = %0.4g / %d = %0.4g\n", stat, npts-nvpars, stat/(npts-nvpars) );
    return s;
}

add_slang_statistic( "myml", &myml_stat, &myml_report; delta_is_chisqr ) ; 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



-- Dave

David Huenemoerder  617-253-4283 (o); -253-8084 (f); http://space.mit.edu/home/dph
MIT Kavli Institute for Astrophysics and Space Research
One Hampshire Street, Room NE80-6065
Cambridge, MA  02139
[Admin. Asst.: Elaine Tirrell, 617-253-7480, egt_at_email.domain.hidden
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