标题： 一个分母为$N$的无偏方差估计量 显示英文标题

标题： An Unbiased Variance Estimator with Denominator $N$

摘要： 标准做法是通过除以$N-1$而不是$N$来获得无偏的方差估计量。然而，如果只使用一半的数据来计算均值，除以$N$仍可以得到无偏估计量。我们证明了另一种均值估计量$\hat{X} = \sum c_n X_n$可以通过分母为$N$来产生这种无偏方差估计量。这些平均调整后的无偏方差（AAUV）允许无穷多种无偏形式，尽管每种形式的方差都比通常的样本方差大。此外，通过对任何 AAUV 进行排列和对称化，可以恢复具有分母$N-1$的经典公式。我们进一步通过在标准均值和基于 AAUV 的均值之间插值展示了连续的无偏方差。将这种平均调整方法扩展到更高阶矩仍然是未来研究的一个课题。 显示英文摘要

摘要： Standard practice obtains an unbiased variance estimator by dividing by $N-1$ rather than $N$. Yet if only half the data are used to compute the mean, dividing by $N$ can still yield an unbiased estimator. We show that an alternative mean estimator $\hat{X} = \sum c_n X_n$ can produce such an unbiased variance estimator with denominator $N$. These average-adjusted unbiased variance (AAUV) permit infinitely many unbiased forms, though each has larger variance than the usual sample variance. Moreover, permuting and symmetrizing any AAUV recovers the classical formula with denominator $N-1$. We further demonstrate a continuum of unbiased variances by interpolating between the standard and AAUV-based means. Extending this average-adjusting method to higher-order moments remains a topic for future work.