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

Andrew T. Karl

Abstract

In Individuals and Moving Range (I-MR) charts, the process standard deviation is often estimated by the span-2 average moving range, scaled by the usual constant $d_2$. Unlike the sample standard deviation, this estimator depends on the observation order: permuting the values can change the average moving range. We make this dependence explicit by modeling the order as an independent uniformly random permutation. A direct application of the law of total variance then decomposes its variance into a component due to ordering and a component due to the realized values. Averaging over all permutations yields a simple order-invariant baseline for the moving-range estimator: the sample Gini mean difference divided by $d_2$. Simulations quantify the resulting fraction of variance attributable to ordering under i.i.d. Normal sampling, and two NIST examples illustrate a typical ordering and an ordering with strong serial structure relative to random permutations of the same values.

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

Abstract

In Individuals and Moving Range (I-MR) charts, the process standard deviation is often estimated by the span-2 average moving range, scaled by the usual constant . Unlike the sample standard deviation, this estimator depends on the observation order: permuting the values can change the average moving range. We make this dependence explicit by modeling the order as an independent uniformly random permutation. A direct application of the law of total variance then decomposes its variance into a component due to ordering and a component due to the realized values. Averaging over all permutations yields a simple order-invariant baseline for the moving-range estimator: the sample Gini mean difference divided by . Simulations quantify the resulting fraction of variance attributable to ordering under i.i.d. Normal sampling, and two NIST examples illustrate a typical ordering and an ordering with strong serial structure relative to random permutations of the same values.
Paper Structure (17 sections, 16 equations, 4 figures, 1 table)

This paper contains 17 sections, 16 equations, 4 figures, 1 table.

Table of Contents

  1. Background and motivation
  2. Setup and the moving-range functional
  3. A law of total variance decomposition
  4. Variance split into values and ordering components
  5. Permutation mean equals a Gini baseline
  6. Closed-form under Normal sampling
  7. Variance of the permutation mean.
  8. Variance of the observed-order MR(2) estimator $T(X,\Pi)$.
  9. Limiting order fraction and a comparison point
  10. Simulation study
  11. Illustrations with NIST data
  12. A typical Individuals-chart example
  13. An order-structured time series
  14. How to use this decomposition in practice
  15. Discussion and conclusion
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Check of the law of total variance identity \ref{['eq:lotv']} under i.i.d. $N(0,1)$ sampling for $n\in\{4,8,12,16,20,25,50\}$. Red circles show the Monte Carlo estimate of the total variance $\widehat{\mathrm{Var}}\{T(X,\Pi)\}$ across $R$ simulated data sets. Blue squares show the Monte Carlo reconstruction $\widehat{\mathbb{E}\{\mathrm{Var}(T\mid X)\}}+\widehat{\mathrm{Var}\{\bar{T}(X)\}}$, where $\mathrm{Var}(T\mid X)$ is approximated using $B$ random permutations per data set and $\bar{T}(X)=\mathbb{E}_{\Pi}\{T(X,\Pi)\mid X\}$ is the permutation mean. The black curve is the exact Normal-reference $\mathrm{Var}\{T(X,\Pi)\}$ from \ref{['eq:VarT_normal_sigma']} with $\sigma^2=1$. Here $R=50{,}000$ and $B=3{,}000$.
  • Figure 2: Order fraction \ref{['eq:orderfraction']} under i.i.d. $N(0,1)$ sampling, shown on a zoomed vertical scale. Green circles give the Monte Carlo estimate of $\mathrm{OrderFraction}(n)=\mathbb{E}\{\mathrm{Var}(T\mid X)\}/\mathrm{Var}\{T(X,\Pi)\}$ using $R$ simulated data sets and $B$ random permutations per data set to approximate $\mathrm{Var}(T\mid X)$. The black curve is the exact Normal-reference $\mathrm{OrderFraction}(n)$ obtained by substituting \ref{['eq:VarTbar_normal_sigma']} and \ref{['eq:VarT_normal_sigma']} into \ref{['eq:orderfraction_equiv']}. The horizontal dashed line marks the limiting value $\mathrm{OrderFraction}(\infty)\approx 0.3813$ from \ref{['eq:OrderFraction_limit_normal']}. Here $R=50{,}000$ and $B=3{,}000$.
  • Figure 3: Permutation distribution of $T(X,\Pi)$ for the NIST Individuals-chart flowrate illustration (first $n=10$ batches) NIST_imr. The histogram shows $\{T(X,\Pi_b)\}_{b=1}^{B}$ for $B=50{,}000$ random permutations of the observed values. Black dashed line: the permutation mean $\bar{T}(X)=\mathbb{E}_{\Pi}\{T(X,\Pi)\mid X\}$ (see \ref{['eq:gmd']}). Red line: the observed-order estimate $T_{\mathrm{obs}}=T(X,\Pi_{\mathrm{obs}})$.
  • Figure 4: Permutation distribution of $T(X,\Pi)$ for the NIST/SEMATECH dataset F_SERIES.DAT (70 consecutive yields) NIST_fseries. The histogram shows $\{T(X,\Pi_b)\}_{b=1}^{B}$ for $B=50{,}000$ random permutations of the observed values. Black dashed line: the permutation mean $\bar{T}(X)=\mathbb{E}_{\Pi}\{T(X,\Pi)\mid X\}$ (see \ref{['eq:gmd']}). Red line: the observed-order estimate $T_{\mathrm{obs}}=T(X,\Pi_{\mathrm{obs}})$.