Computation of the (zero-state) Average Run Length (ARL) for EWMA control charts (based on the sample variance \(S^2\)) monitoring normal variance with estimated parameters.
sewma.arl.prerun(l, cl, cu, sigma, df1, df2, hs=1, sided="upper",
r=40, qm=30, qm.sigma=30, truncate=1e-10)Returns a single value which resembles the ARL.
smoothing parameter lambda of the EWMA control chart.
lower control limit of the EWMA control chart.
upper control limit of the EWMA control chart.
true standard deviation.
actual degrees of freedom, corresponds to subgroup size (for known mean it is equal to the subgroup size, for unknown mean it is equal to subgroup size minus one.
degrees of freedom of the pre-run variance estimator.
so-called headstart (enables fast initial response).
distinguishes between one- and two-sided two-sided EWMA-\(S^2\) control charts
by choosing "upper" (upper chart without reflection at cl -- the actual value of
cl is not used), "Rupper" (upper chart with reflection at cl),"Rlower"
(lower chart with reflection at cu),
and "two" (two-sided chart), respectively.
dimension of the resulting linear equation system (highest order of the collocation polynomials).
number of quadrature nodes for calculating the collocation definite integrals.
number of quadrature nodes for convoluting the standard deviation uncertainty.
size of truncated tail.
Sven Knoth
Essentially, the ARL function sewma.arl is convoluted with the
distribution of the sample standard deviation.
For details see Jones/Champ/Rigdon (2001) and Knoth (2014?).
L. A. Jones, C. W. Champ, S. E. Rigdon (2001), The performance of exponentially weighted moving average charts with estimated parameters, Technometrics 43, 156-167.
S. Knoth (2005), Accurate ARL computation for EWMA-\(S^2\) control charts, Statistics and Computing 15, 341-352.
S. Knoth (2006), Computation of the ARL for CUSUM-\(S^2\) schemes, Computational Statistics & Data Analysis 51, 499-512.
sewma.arl for zero-state ARL function of EWMA control charts w/o pre run uncertainty.