Computation of quantiles of the Run Length (RL) for EWMA control charts monitoring normal variance.
sewma.q.prerun(l,cl,cu,sigma,df1,df2,alpha,hs=1,sided="upper",
r=40,qm=30,qm.sigma=30,truncate=1e-10)sewma.q.crit.prerun(l,L0,alpha,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 a single value which resembles the RL quantile of order alpha and
the lower and upper control limit cl and cu, respectively.
smoothing parameter lambda of the EWMA control chart.
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.
true and in-control standard deviation, respectively.
in-control quantile value.
quantile level.
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.
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
Instead of the popular ARL (Average Run Length) quantiles of the EWMA
stopping time (Run Length) are determined. The algorithm is based on
Waldmann's survival function iteration procedure.
Thereby the ideas presented in Knoth (2007) are used.
sewma.q.crit.prerun determines the critical values (similar to alarm limits)
for given in-control RL quantile L0 at level alpha by applying secant
rule and using sewma.sf().
In case of sided="two" and mode="unbiased"
a two-dimensional secant rule is applied that also ensures that the
minimum of the cdf for given standard deviation is attained at sigma0.
S. Knoth (2007), Accurate ARL calculation for EWMA control charts monitoring simultaneously normal mean and variance, Sequential Analysis 26, 251-264.
K.-H. Waldmann (1986), Bounds for the distribution of the run length of geometric moving average charts, Appl. Statist. 35, 151-158.
sewma.q and sewma.q.crit for the version w/o pre-run uncertainty.