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

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