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

The 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 χ2 statistic, is its applicability without the need to combine data points. This feature has made the 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 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 statistic by studying (a) the distribution of C statistic for a fully specified model, (b) the distribution of C_min 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 ΔC statistic that is used for parameter estimation. The results confirm the expectation that, in the high-count limit, both C statistic and C_min have the same mean and variance as a χ2 statistic with same number of degrees of freedom. It is also found that, in the low-count regime, the expectation of the C statistic and C_min can be substantially lower than for a 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 statistic to Poisson data.