A comparison of methods for Poisson regression in the presence of background

TL;DR

This paper compares three Poisson regression methods with background, finding the joint-fit approach most reliable and unbiased, while non-parametric and fixed-background methods can be biased, especially in low-count regimes.

Contribution

It provides a detailed statistical comparison of Poisson regression methods with background, highlighting the advantages of the joint-fit approach over others.

Findings

Non-parametric background method is significantly biased in low-count regimes.

Joint-fit method allows reliable hypothesis testing and unbiased parameter estimation.

Non-parametric method adds more degrees of freedom than justified by the model.

Abstract

This paper provides a statistical analysis of three common methods of regression for Poisson data in the presence of Poisson background, namely the joint fit with two parametric models for the source and the background, the use of a non-parametric model for the background known as the wstat method, and the regression with a fixed background. The non-parametric background method, which is a popular method for spectral data, is found to be significantly biased, especially in the low-count and background-dominated regimes. Similar conclusions apply to the fixed-background regression. The joint-fit method, on the other hand, simultaneously affords reliable hypothesis testing by means of the usual Cash statistic and unbiased reconstruction of source parameters. We also investigate the effect of non-parametric regression on the number of effective degrees of freedom by means of the Efron degree of freedom function. We find that the wstat method adds a significantly larger number of degrees of freedom, compared to the number of free parameters in the source model. The other two methods have a number of degrees of freedom consistent with the number of adjustable parameters, at least for the simple models investigated in this paper.