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

2020/09/16 by Bonamente, Massimiliano, Spence, David
#FOS: Computer and information sciences #FOS: Physical sciences #Instrumentation and Methods for Astrophysics (astro-ph.IM) #Methodology (stat.ME)

paper · doi:10.48550/arxiv.2009.07915

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 χ² statistic.

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