Parameter Estimation in Astronomy with Poisson-Distributed Data. II. The   Modified Chi-Square-Gamma Statistic
  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) for 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 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 values can be calculated with the incomplete gamma function. I give a practical demonstration showing how the modified chi-square-gamma statistic can be used in experimental astrophysics by analyzing simulated X-ray observations of a weak point source (signal-to-noise ratio of 5.2; 40 photons spread over 317 pixels) on a noisy background (0.06 photons per pixel). Accurate estimates (95% confidence intervals/limits) of the location and intensity of the X-ray point source are determined. 
Kenneth J. Mighell (1) ((1) National Optical Astronomy Observatory)