Multinomial, Poisson and Gaussian statistics in count data analysis

TL;DR

This paper demonstrates that Gaussian approximation is inadequate for count data analysis in neutron scattering, proposing Multinomial and Poisson methods for more accurate, unbiased results, especially at low counts.

Contribution

It introduces a Multinomial approach for count data analysis, comparing it with Poisson and Gaussian methods, and shows its advantages in neutron scattering experiments.

Findings

Multinomial method provides nearly unbiased results.

Gaussian approximation is biased but more robust in some cases.

Multinomial can outperform Poisson in certain scenarios.

Abstract

It is generally known that counting statistics is not correctly described by a Gaussian approximation. Nevertheless, in neutron scattering, it is common practice to apply this approximation to the counting statistics; also at low counting numbers. We show that the application of this approximation leads to skewed results not only for low-count features, such as background level estimation, but also for its estimation at double-digit count numbers. In effect, this approximation is shown to be imprecise on all levels of count. Instead, a Multinomial approach is introduced as well as a more standard Poisson method, which we compare with the Gaussian case. These two methods originate from a proper analysis of a multi-detector setup and a standard triple axis instrument.We devise a simple mathematical procedure to produce unbiased fits using the Multinomial distribution and demonstrate this method on synthetic and actual inelastic scattering data. We find that the Multinomial method provide almost unbiased results, and in some cases outperforms the Poisson statistics. Although significantly biased, the Gaussian approach is in general more robust in cases where the fitted model is not a true representation of reality. For this reason, a proper data analysis toolbox for low-count neutron scattering should therefore contain more than one model for counting statistics.