  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.
