Poisson vs. Gaussian statistics for sparse X-ray data: Application to the soft X-ray spectrometer

Reliable results when fitting X-ray data require proper consideration of the statistics involved. We probe the impact of Gaussian versus Poisson statistics at low count levels using both the standard chi(2) method and maximum likelihood based on Poisson (C) statistics. The difference is studied and quantified through simulated spectra with known properties. We then test the results through analysis of Mn K alpha calibration data taken with the flight spare microcalorimeter for the Hitomi soft X-ray spectrometer. Through comparison with simulations, our results show that the chi(2) method tends to give overly optimistic estimates of the detector energy resolution, in particular when there are few counts. Given an energy resolution of similar to 5eV and a line with about 100 photons, the line width becomes similar to 10% lower in the chi(2) method than in Poisson statistics. This is a consequence of the uncertainties being dominated by counting statistics, and therefore highlights the need to choose the appropriate fit statistic.