Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912957217112064 |
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| author | Karl, Andrew T. |
| author_facet | Karl, Andrew T. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_20007 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition Karl, Andrew T. Statistics Theory Methodology 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. |
| title | Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2602.20007 |