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.

翻译：本文对存在泊松本底时三种常见泊松数据回归方法进行了统计分析，即：采用源与本底的双参数模型联合拟合、使用称为wstat方法的非参数本底模型，以及固定本底回归。研究发现，非参数本底方法（一种常用于谱数据的方法）存在显著偏差，尤其是在低计数和本底主导区域。类似结论也适用于固定本底回归。相比之下，联合拟合方法既能通过常规Cash统计量实现可靠的假设检验，又能无偏重建源参数。我们进一步借助Efron自由度函数研究了非参数回归对有效自由度数量的影响。结果表明，与源模型中的自由参数数量相比，wstat方法引入了显著更多的自由度。而另外两种方法的自由度数量与可调参数数量一致——至少对于本文研究的简单模型而言如此。