On the use of C-stat in testing models for X-ray spectra

TL;DR

This paper demonstrates that the C-statistic can be effectively used for goodness-of-fit testing in X-ray spectral analysis by providing formulas for its expected value and variance, enabling reliable statistical tests even with low-count spectra.

Contribution

It introduces methods to calculate the expected value and variance of the C-statistic, allowing its use in statistical goodness-of-fit tests for X-ray spectra.

Findings

C-statistic can be used for goodness-of-fit testing.

Formulas for expected value and variance of C-statistic are provided.

Reliable tests are possible with spectra having ~30 counts.

Abstract

It has been shown that for the analysis of X-ray spectra the C-statistic, contrary to the chi^2-statistic, provides unbiased estimates of the model parameters and their uncertainty ranges. However, it is often stated that the C-statistic cannot be used to carry out statistical tests on the goodness of fit of the model, and therefore several investigations are still based on chi^2-statistics. Here we show that it is straightforward to calculate the expected value and variance of the C-statistic so that it can be used in tests. We provide formulae and simple numerical approximations to evaluate these expected values and variances. We also give examples indicating that tests based on only the expected value and variance of the C-statistic are reliable for spectra even with only ~30 counts. Therefore the C-statistic can be used for statistical tests such as assessing the goodness of fit of a spectral model.