Distribution of the C statistic with applications to the sample mean of Poisson data

The ${C} $C statistic, also known as the Cash statistic, is often used in astronomy for the analysis of low-count Poisson data. The main advantage of this statistic, compared to the more commonly used $\chi ^2 $χ2 statistic, is its applicability without the need to combine data points. This feature has made the ${C} $C statistic a very useful method to analyze Poisson data that have small (or even null) counts in each resolution element. One of the challenges of the ${C} $C statistic is that its probability distribution, under the null hypothesis that the data follow a parent model, is not known exactly. This paper presents an effort towards improving our understanding of the ${C} $C statistic by studying (a) the distribution of ${C} $C statistic for a fully specified model, (b) the distribution of Cmin resulting from a maximum-likelihood fit to a simple one-parameter constant model, i.e. a model that represents the sample mean of N Poisson measurements, and (c) the distribution of the associated $\Delta C $ΔC statistic that is used for parameter estimation. The results confirm the expectation that, in the high-count limit, both ${C} $C statistic and Cmin have the same mean and variance as a $\chi ^2 $χ2 statistic with same number of degrees of freedom. It is also found that, in the low-count regime, the expectation of the ${C} $C statistic and Cmin can be substantially lower than for a $\chi ^2 $χ2 distribution. The paper makes use of recent X-ray observations of the astronomical source PG 1116+215 to illustrate the application of the ${C} $C statistic to Poisson data.