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

The $$ statistic, also known as the Cash statistic, is often used in astronomy for the analysis of low-count Poisson data. The main advantage of this statistic, compared to the more commonly used $$ statistic, is its applicability without the need to combine data points. This feature has made the $$ statistic a very useful method to analyze Poisson data that have small (or even null) counts in each resolution element. One of the challenges of the $$ 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 $$ statistic by studying (a) the distribution of $$ 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 $$ statistic that is used for parameter estimation. The results confirm the expectation that, in the high-count limit, both $$ statistic and C _min have the same mean and variance as a $$ statistic with same number of degrees of freedom. It is also found that, in the low-count regime, the expectation of the $$ statistic and C _min can be substantially lower than for a $$ distribution. The paper makes use of recent X-ray observations of the astronomical source PG 1116+215 to illustrate the application of the $$ statistic to Poisson data.