多分布中微子截面测量中共享事件的拟合优度检验
Goodness-of-fit for multi-distribution neutrino cross-section measurements with shared events
AI总结：
针对共享事件导致的多分布中微子截面测量协方差矩阵秩亏问题，提出范围投影卡方统计量，通过将检验限制在独立信息子空间，得到有效自由度并验证其有效性。
AI中文摘要：
当从共同的事件样本中提取多个微分截面测量时，同一事件会同时贡献给多个分布。这种事件共享结构在箱计数之间施加了精确的线性约束，将测量的有效维度降低到总箱数以下。由于协方差矩阵遵循这些分布间约束，它存在秩亏。虽然有限的数值精度可能无意中恢复可逆性，但沿约束维度的$\chi^{2}$贡献将是任意的，从而在全球拟合优度检验中产生非物理的膨胀$\chi^{2}$值。我们提出了范围投影$\chi^{2}$，这是一种检验统计量，将拟合优度检验限制在携带独立统计信息的子空间上，产生具有$N_\text{bins} - N_\text{null}$自由度的$\chi^{2}$，其中$N_\text{null}$是独立约束的数量。我们表明，这种秩亏是事件共享结构的可预测结果，并且$N_\text{null}$分解为先验地由分箱几何确定的结构贡献和依赖于相空间占据的运动学贡献。该方法通过解析玩具模型和模拟的中微子-氩截面测量（包括去折叠和多源系统不确定性）进行了验证。
英文摘要：
When multiple differential cross-section measurements are extracted from a common event sample, the same events contribute simultaneously to multiple distributions. This event-sharing structure imposes exact linear constraints among the bin counts, reducing the effective dimensionality of the measurement below the total number of bins. Since the covariance matrix obeys these inter-distribution constraints, it contains a rank deficiency. While limited numerical precision may inadvertently restore invertibility, the $χ^{2}$ contributions along constrained dimensions will be arbitrary, yielding unphysically inflated $χ^{2}$ values in global goodness-of-fit tests. We present the range-projected $χ^{2}$, a test statistic that restricts the goodness-of-fit test to the subspace carrying independent statistical information, yielding a $χ^{2}$ with $N_\text{bins} - N_\text{null}$ degrees of freedom, where $N_\text{null}$ is the number of independent constraints. We show that this rank deficiency is a predictable consequence of the event-sharing structure, and that $N_\text{null}$ decomposes into a structural contribution determined a priori from the binning geometry and a kinematic contribution that depends on the phase-space occupancy. The method is validated with an analytical toy model and a simulated neutrino--argon cross-section measurement including unfolding and multi-source systematic uncertainties.
