Computation of quantiles of the Run Length (RL) for EWMA control charts monitoring normal variance.
sewma.crit.prerun(l,L0,df1,df2,sigma0=1,cl=NULL,cu=NULL,hs=1,sided="upper",
mode="fixed",r=40,qm=30,qm.sigma=30,truncate=1e-10,
tail_approx=TRUE,c.error=1e-10,a.error=1e-9)Returns the lower and upper control limit cl and cu.
smoothing parameter lambda of the EWMA control chart.
in-control quantile value.
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.
true and in-control standard deviation, respectively.
deployed for sided="Rupper", that is, upper variance control chart with lower
reflecting barrier cl.
for two-sided (sided="two") and fixed upper control limit
(mode="fixed") a value larger than sigma0
has to been given, for all other cases cu is ignored.
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.
only deployed for sided="two" -- with "fixed" an upper control limit
(see cu) is set and only the lower is calculated to obtain the in-control ARL L0, while
with "unbiased" a certain unbiasedness of the ARL function is guaranteed (here, both the
lower and the upper control limit are calculated).
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.
controls whether the geometric tail approximation is used (is faster) or not.
error bound for two succeeding values of the critical value during applying the secant rule.
error bound for the quantile level alpha during applying the secant rule.
Sven Knoth
sewma.crit.prerun determines the critical values (similar to alarm limits)
for given in-control ARL L0
by applying secant rule and using sewma.arl.prerun().
In case of sided="two" and mode="unbiased"
a two-dimensional secant rule is applied that also ensures that the
maximum of the ARL function for given standard deviation is attained
at sigma0. See Knoth (2010) for some details of the algorithm involved.
H.-J. Mittag and D. Stemann and B. Tewes (1998), EWMA-Karten zur \"Uberwachung der Streuung von Qualit\"atsmerkmalen, Allgemeines Statistisches Archiv 82, 327-338, S. Knoth (2005), Accurate ARL computation for EWMA-\(S^2\) control charts, Statistics and Computing 15, 341-352.
S. Knoth (2010), Control Charting Normal Variance -- Reflections, Curiosities, and Recommendations, in Frontiers in Statistical Quality Control 9, H.-J. Lenz and P.-T. Wilrich (Eds.), Physica Verlag, Heidelberg, Germany, 3-18.
sewma.arl.prerun for calculation of ARL of variance charts under
pre-run uncertainty and sewma.crit for
the algorithm w/o pre-run uncertainty.