The Cash statistic, also known as the $C$ statistic, 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.

这 $C$ 灰分统计，也称为 $\mathrm{<span\; data-immersive-translate-walked="013d97f0-4c8f-4881-b545-63d39255f976">C</span>}$统计量，通常用于分析低计数泊松数据，包括自变量的某些值具有空计数的数据。对于无法在不损失分辨率的情况下组合或重新分箱的低计数数据，使用此统计数据尤其有吸引力。本文提出了一种使用基于泊松的现金统计量的线性模型最佳拟合参数的新最大似然解。本文提出的解决方案提供了一种新的简单方法来测量任何基于泊松的数据（包括具有空计数的数据）的线性模型的最佳拟合参数。特别是，该方法强制要求最佳拟合线性模型在自变量的支持范围内必须为非负。该方法概括为一种简单的算法，可以用线性模型拟合任意大小和计数率的泊松计数数据，完全绕过传统的使用 ${\mathrm{<span\; data-immersive-translate-walked="013d97f0-4c8f-4881-b545-63d39255f976">\chi </span>}}^{\mathrm{<span\; data-immersive-translate-walked="013d97f0-4c8f-4881-b545-63d39255f976">2</span>}}$统计。