顺序诱导的移动范围西格玛估计器方差：总方差分解
Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition
AI总结：
本文通过总方差分解揭示了移动范围西格玛估计器中顺序诱导的方差成分，证明了排列均值与吉尼均差的关系，并在正态分布下推导了相邻分量的闭式表达式。
摘要：
I-MR控制图通常通过将跨度2的平均移动范围除以无偏常数d₂来估计过程标准差σ；与无偏样本标准差（S/c₄）不同，这个估计器依赖于顺序通过相邻关系，因此固定样本的排列变化会改变它。我们通过引入一个独立的均匀随机排列并应用总方差定律，得到一个精确的分解，分为一个值分量（排列均值的方差）和一个相邻分量（排列下的条件方差的期望）。排列均值是顺序不变的，并等于GMD/d₂，其中GMD是样本吉尼均差。在i.i.d.正态抽样下，两个分量都具有闭式表达式；相邻分量的分数收敛到0.3813，而相对于S/c₄的已知渐近效率损失几乎完全源于相邻效应。
英文摘要：
I--MR charts commonly estimate the process standard deviation $σ$ via the span-2 average moving range divided by the unbiasing constant $d_2$; unlike the unbiased sample standard deviation ($S/c_4$), this estimator depends on ordering through adjacency, so permuting a fixed sample changes it. We formalize this by introducing an independent uniformly random permutation and applying the law of total variance, yielding an exact decomposition into a values component (variance of the permutation mean) and an adjacency component (expected conditional variance over permutations). The permutation mean is order-invariant and equals $\GMD/d_2$, where $\GMD$ is the sample Gini mean difference. Under i.i.d.\ Normal sampling, both components admit closed forms; the adjacency fraction converges to $0.3813$, and the familiar asymptotic efficiency loss relative to $S/c_4$ is almost entirely an adjacency effect.
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