A semi-analytical solution to the maximum likelihood fit of Poisson data to a linear model using the Cash statistic

TL;DR

This paper introduces a semi-analytical maximum likelihood method for fitting a linear model to low-count Poisson data using the Cash statistic, ensuring non-negativity and avoiding traditional chi-squared methods.

Contribution

It presents a novel, simple algorithm for fitting Poisson data with a linear model using the Cash statistic, applicable to data with null counts and enforcing non-negativity.

Findings

Provides a semi-analytical solution for maximum likelihood fitting

Enforces non-negativity of the linear model across the support

Bypasses traditional chi-squared fitting methods

Abstract

[ABRIDGED] The Cash statistic, also known as the C stat, is commonly used for the analysis of low-count Poisson data, including data with null counts for certain values of the independent variable. The use of this statistic is especially attractive for low-count data that cannot be combined, or re-binned, without loss of resolution. This paper presents a new maximum-likelihood solution for the best-fit parameters of a linear model using the Poisson-based Cash statistic. The solution presented in this paper provides a new and simple method to measure the best-fit parameters of a linear model for any Poisson-based data, including data with null counts. In particular, the method enforces the requirement that the best-fit linear model be non-negative throughout the support of the independent variable. The method is summarized in a simple algorithm to fit Poisson counting data of any size and counting rate with a linear model, by-passing entirely the use of the traditional $\chi^2$ statistic.