Kenneth J. Mighell

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-square 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-square 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 work 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 (LTSA) program (NRA 95-OSS-16).
Kenneth Mighell
Associate Scientist  
Kitt Peak National Observatory
National Optical Astronomy Observatories
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Last updated: 2000 April 5