GOODNESS-OF-FIT TESTING OF LOW-COUNT DATA USING THE MODIFIED CHI-SQUARE-GAMMA STATISTIC
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.
Kenneth Mighell Associate Scientist Kitt Peak National Observatory National Optical Astronomy Observatories EMAIL: firstname.lastname@example.org MAIL: P.O. Box 26732, Tucson, AZ 85726-6732 FEDEX: 950 N. Cherry Ave., Tucson, AZ 85719 PHONE: (520) 318-8391 FAX: (520) 318-8360 URL: http://www.noao.edu/staff/mighell/