让高阶渐近方法实用化：校正天文计数数据的拟合优度检验
Making high-order asymptotics practical: correcting goodness-of-fit test for astronomical count data
AI总结：
针对低计数Poisson天文数据中C统计量拟合优度检验失效的问题，本文提出基于条件化与高阶渐近的校正算法和校准p值，证明其精度更高且计算成本低。
摘要：
C统计量是一种广泛用于高能物理和天体物理学中Poisson数据模型拟合与拟合优度评估的似然比统计量。尽管它具有便利的渐近性质，但该统计量常被应用于其名义零分布依赖于无根据假设的情形。由于研究人员通常不会进行稳健性检查，他们的科学发现容易受到误导性显著性计算的影响。本文以低计数情形为重点，全面研究了C统计量及相关拟合优度算法的理论性质。我们聚焦于常见的“代入（plug-in）”算法，这类算法通过假定真实参数等于其估计值来获得C的矩。为校正这些方法，我们提供了一套新的、有原则依据且便于用户使用的算法以及校准良好的p值，可立即部署到（天体）物理数据分析流程中。结合理论与数值结果，我们表明：(a) 标准的基于χ²的拟合优度评估在低计数情形下无效；(b) 朴素方法（例如普通bootstrap）会产生有偏的零分布；(c) 基于条件化和高阶渐近的校正Z检验能以较低计算成本给出最佳精度。我们通过一系列模拟和实际天体物理分析说明了所提方法，并在GitHub仓库中提供了开源Python包。
英文摘要：
The C statistic is a widely used likelihood-ratio statistic for model fitting and goodness-of-fit assessments with Poisson data in high-energy physics and astrophysics. Although it enjoys convenient asymptotic properties, the statistic is routinely applied in cases where its nominal null distribution relies on unwarranted assumptions. Because researchers do not typically carry out robustness checks, their scientific findings are left vulnerable to misleading significance calculations. With an emphasis on low-count scenarios, we present a comprehensive study of the theoretical properties of C statistics and related goodness-of-fit algorithms. We focus on common ``plug-in'' algorithms where moments of C are obtained by assuming the true parameter equals its estimate. To correct such methods, we provide a suite of new principled user-friendly algorithms and well-calibrated p-values that are ready for immediate deployment in the (astro)physics data-analysis pipeline. Using both theoretical and numerical results, we show (a) standard $χ^2$-based goodness-of-fit assessments are invalid in low-count settings, (b) naive methods (e.g., vanilla bootstrap) result in biased null distributions, and (c) the corrected Z-test based on conditioning and high-order asymptotics gives the best precision with low computational cost. We illustrate our methods via a suite of simulations and applied astrophysical analyses. An open-source Python package is provided in a GitHub repository.
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