**AAS 196th Meeting, June 2000**

*Session 54. Applications of Statistics*

Special Contributed Display, Thursday, June 8, 2000, 9:20am-4:00pm, Empire Hall South
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## [54.05] Goodness-of-fit Testing of Low-Count Data using the Modified Chi-Square-Gamma Statistic

*K. Mighell (KPNO/NOAO)*

I investigate the use of Pearson's chi-square statistic, the
Maximum Likelihood Ratio statistic for Poisson
distributions, and the chi-square-gamma statistic (Mighell
1999, ApJ, 518, 380) in the determination of the
goodness-of-fit between theoretical models and low-count
Poisson-distributed data. I demonstrate that these
statistics should not be used to determine the
goodness-of-fit with data values of 10 or less.

I modify the chi-square-gamma statistic for the purpose of
improving its goodness-of-fit performance. I demonstrate
that the modified chi-square-gamma statistic performs
(nearly) like an ideal \chi^{2} statistic for the
determination of goodness-of-fit with low-count data. On
average, for the correct (true) models, the mean value of
modified chi-square-gamma statistic is equal to the number
of degrees of freedom (\nu) and its variance is 2\nu
like the \chi^{2} distribution for \nu degrees of freedom.
Probabilities for modified chi-square-gamma goodness-of-fit
determinations can be made using the incomplete gamma
function.

Simulated X-ray observations of a background flux of 0.06
photons per pixel and a point source of 40 photons spread
over 317 pixels are analyzed as a practical demonstration of
the use of the modified chi-square-gamma statistic in
experimental astrophysics.

This research is supported by a grant from the National
Aeronautics and Space Administration (NASA), Order No.\
S-67046-F, which was awarded by the Long-Term Space
Astrophysics Program (NRA 95-OSS-16).

If you would like more information about this abstract, please
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