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The C statistics, also known as the Cash statistic, is often used inastronomy for the analysis of low-count Poisson data. One of the challenges ofthe C statistic is that its probability distribution, under the null hypothesisthat the data follow a parent model, is not known exactly. Such distribution isneeded for model testing, namely to determine the acceptability of models andthen to determine confidence intervals of model parameters. This is in contrastwith the accurate knowledge, for Gaussian data, of the $\chi^2$ statistic forany number of free parameters in the parent model.  This paper presents an effort towards improving our understanding of the Cstatistic by studying (a) the distribution of for a fully specified model, (b)the distribution of $C_{min}$ resulting from a maximum-likelihood fit to asimple one-parameter constant model, i.e., a model that represents the samplemean of $N$ Poisson measurements, and (c) the distribution of the associated$\Delta C$ statistic that is used for parameter estimation. The results confirmthe expectation that, in the high-count limit, both statistics have the samemean and variance as a $\chi^2$ statistic with same number of degrees offreedom. It is also found that, in the low-count regime, the expectation of theC statistic can be substantially lower than for a $\chi^2$ distribution. Theseresult have implications for hypothesis testing in the low-count Poisson regimethat are also discussed in the paper.  The paper makes use of recent X-ray observations of the astronomical sourcePG 1116+215 to illustrate the application of the C statistic to Poisson data.These measurements are also used to identify biases in the use of the $\chi^2$statistic for Poisson data, especially in the low-count regime.

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