Analytic formulas to calculate statistical uncertainties of $v_n\left\{2k\right\}$ harmonics extracted from the Q cumulants are presented. The Q cumulants are multivariate polynomial functions of the weighted means of $2m$-particle azimuthal correlations, $\langle \langle 2m\rangle \rangle$. Variances and covariances of the $\langle \langle 2m\rangle \rangle$ are included in the analytic formulas of the uncertainties that can be calculated simultaneously with the calculations of the $v_n\left\{2k\right\}$ harmonics. The calculations are performed using a simple toy model, which roughly simulates elliptic flow azimuthal anisotropy with magnitudes around 0.05. The results are compared with the results obtained by the many data subsets, and by the bootstrapping method. The first one is a common way of estimation of the statistical uncertainties of the $v_n\left\{2k\right\}$ harmonics in a real experiment. In order to increase precision in the measurement of the $v_n\left\{2k\right\}$ harmonics, a large number of 15 000 subsets and the same number of the resampling in the bootstrap method is used. Unlike the other ways of the analytic calculation of the $v_n\left\{2k\right\}$ statistical uncertainties, our proposal that includes the use of squared weights in the calculation of both the variances and covariances, gives the best agreement with the results obtained from the subsets and bootstrap method. Additionally, a recurrence equation between Q cumulants of any order is also presented.

中文翻译：

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来自 Qcumulants 的 vn{2k} 谐波的统计不确定性

计算统计不确定度的解析公式 $v_n\left\{2克\right\}$呈现从 Q 累积量中提取的谐波。Q 累积量是加权均值的多元多项式函数$2米$-粒子方位角相关性， $〈〈2米〉〉$. 的方差和协方差$〈〈2米〉〉$ 包含在不确定性的分析公式中，可以与计算的同时计算 $v_n\left\{2克\right\}$谐波。计算是使用一个简单的玩具模型进行的，该模型粗略地模拟了 0.05 左右的椭圆流方位各向异性。将结果与通过许多数据子集和自举方法获得的结果进行比较。第一种是估计统计不确定性的常用方法$v_n\left\{2克\right\}$实际实验中的谐波。为了提高测量精度$v_n\left\{2克\right\}$谐波，大量的 15 000 个子集和相同数量的自举方法中的重采样被使用。与其他解析计算方法不同的是$v_n\left\{2克\right\}$统计不确定性，我们的建议包括在方差和协方差的计算中使用平方权重，与从子集和引导方法获得的结果最一致。此外，还给出了任意阶 Q 累积量之间的递推方程。