Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition

TL;DR

This paper analyzes how the ordering of data affects the moving-range sigma estimator in control charts, revealing that adjacency significantly influences its variance and efficiency.

Contribution

It introduces a total-variance decomposition for the order-dependent moving-range estimator using permutation theory and provides closed-form results under normal sampling.

Findings

Permutation mean equals Gini mean difference divided by d_2

Adjacency component converges to approximately 0.3813

Efficiency loss is mainly due to adjacency effects

Abstract

I--MR charts commonly estimate the process standard deviation $\sigma$ 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.